# Tool Unlocked: Equilateral Triangles

June 20, 2014 9:38 AM Subscribe

Nifty, but bad at recognizing valid solutions that are different from the hardcoded answers. You will likely run into this at Level 5.

posted by ryanrs at 10:00 AM on June 20, 2014 [3 favorites]

posted by ryanrs at 10:00 AM on June 20, 2014 [3 favorites]

This game is for squares.

posted by fairmettle at 10:00 AM on June 20, 2014 [2 favorites]

posted by fairmettle at 10:00 AM on June 20, 2014 [2 favorites]

I love that the comments on the site have at least two instances of:

-My solution isn't recognised, you have a bug!

--Oops, never mind. Solved it!

posted by metaBugs at 10:03 AM on June 20, 2014

-My solution isn't recognised, you have a bug!

--Oops, never mind. Solved it!

posted by metaBugs at 10:03 AM on June 20, 2014

*You will likely run into this at Level 5.*I ran into it at level 3. i'm not used to thinking that I can have an equilateral triangle available at the push of a button so I think about everything as a compass and straight edge problem.

posted by Dr. Twist at 10:14 AM on June 20, 2014 [1 favorite]

Pre-Taped Call In Show: "

and also I got depression

posted by boo_radley at 10:14 AM on June 20, 2014 [3 favorites]

*I am even wearing my geometry t-shirt*"and also I got depression

posted by boo_radley at 10:14 AM on June 20, 2014 [3 favorites]

I like it! See also Ancient Greek Geometry (previously). The interface is a bit nicer, but you don't get the 'upgrades' of being able to recreate a figure you did once before at the push of a button.

posted by echo target at 10:17 AM on June 20, 2014 [5 favorites]

posted by echo target at 10:17 AM on June 20, 2014 [5 favorites]

*Nifty, but bad at recognizing valid solutions that are different from the hardcoded answers. You will likely run into this at Level 5.*

That is exactly where I ran into the wall. I'm staring at a parallel line, and it's just not accepting my answer.

posted by explosion at 10:23 AM on June 20, 2014

It also doesn't do a good job when two points end up close together. And not just selecting one when you clicked on the other. On the last level it was consistently creating points well away from where I clicked until I went back and rejiggered some arbitrary points to pull things apart more.

posted by eruonna at 10:24 AM on June 20, 2014

posted by eruonna at 10:24 AM on June 20, 2014

There are some (#7) where you explicitly have to draw your line segment between the points identified, too.

posted by boo_radley at 10:30 AM on June 20, 2014 [1 favorite]

posted by boo_radley at 10:30 AM on June 20, 2014 [1 favorite]

Also, Compass #2 just doesn't seem to work a lot of the time. Which is frustrating because I can't see how else to solve level 20....

posted by Pre-Taped Call In Show at 10:53 AM on June 20, 2014

posted by Pre-Taped Call In Show at 10:53 AM on June 20, 2014

I spotted this weirdly somewhat relevant article in the arXiv email earlier today. Still not enough for me not to hate ruler and compass constructions, though.

posted by hoyland at 10:55 AM on June 20, 2014

posted by hoyland at 10:55 AM on June 20, 2014

This is awesome!

posted by Eyebrows McGee at 11:21 AM on June 20, 2014

posted by Eyebrows McGee at 11:21 AM on June 20, 2014

It doesn't seem to work for me? I can't connect the two points at the start. What am I supposed to do? None of the tools seem to be what I need?

posted by JHarris at 11:23 AM on June 20, 2014

posted by JHarris at 11:23 AM on June 20, 2014

An hour disappeared into a vortex today. Why do I ever surf mf before work?

posted by BrotherCaine at 11:25 AM on June 20, 2014

posted by BrotherCaine at 11:25 AM on June 20, 2014

Could you please describe a tool that you need?It doesn't seem to work for me? I can't connect the two points at the start. What am I supposed to do? None of the tools seem to be what I need?

posted by Flunkie at 11:27 AM on June 20, 2014

*It doesn't seem to work for me? I can't connect the two points at the start. What am I supposed to do? None of the tools seem to be what I need?*

the first level?

posted by Dr. Twist at 11:29 AM on June 20, 2014

Finished! Thank god, now I can get on with my life. I'm an hour late to the pub, thanks a lot Euclid.

posted by Pre-Taped Call In Show at 11:35 AM on June 20, 2014

posted by Pre-Taped Call In Show at 11:35 AM on June 20, 2014

Stuck on level 14. I haven't run into problems with the interface or getting it to accept my solutions, so I'm happy about that. I'm also not sure what Compass 2 is for, but I'm sure I'll figure it out eventually. Excellent game!

posted by rebent at 11:54 AM on June 20, 2014

posted by rebent at 11:54 AM on June 20, 2014

huh, spoke too soon I guess. I reasoned that (SPOILERS?!?!??!) the center of the circle must be equidistant from all 3 lines. The point equidistant from two lines is the bisection of the angle. So I bisected two angels and used that as the midpoint for the circle, using one of the bisection x edges as the point for the edge of the circle. It looks like the solution works! But, it's not moving me forward.

posted by rebent at 11:58 AM on June 20, 2014

posted by rebent at 11:58 AM on June 20, 2014

huh, I guess I spoke too soon, again! I was on the right track, i just forgot one aspect!

posted by rebent at 12:00 PM on June 20, 2014

posted by rebent at 12:00 PM on June 20, 2014

Okay, dammit, closing the tab, must get work done. Addictive! Reminds me of when I managed to combine two years of math into one, by working through the proofs of geometry after school with Mr. Buttermore.

posted by tavella at 12:09 PM on June 20, 2014

posted by tavella at 12:09 PM on June 20, 2014

At first I thought that it seemed like you needed to know the stuff already in order to do it. (Like #14, which I remembered)

Now I'm up to #20 and don't know the idea. So I have to figure it out, presumably from stuff I have already done.

Now I think that's the way to teach this stuff.

posted by MtDewd at 12:19 PM on June 20, 2014

Now I'm up to #20 and don't know the idea. So I have to figure it out, presumably from stuff I have already done.

Now I think that's the way to teach this stuff.

posted by MtDewd at 12:19 PM on June 20, 2014

Amazingly, despite taking geometry over 25 years ago, I only had to look things up starting with #14. A fun little game.

posted by jepler at 12:39 PM on June 20, 2014

posted by jepler at 12:39 PM on June 20, 2014

Level 20 has either a lot of almost-solutions that look okay but aren't quite on, or isn't very good at detecting alternate solutions. I thought I got it three times before I followed someone else's directions and finally officially got it.

posted by echo target at 12:41 PM on June 20, 2014

posted by echo target at 12:41 PM on June 20, 2014

I thought my solution to level 11 wasn't being recognized, but while writing up a bug report I realized that BC is not necessarily perpendicular to AC. If something like this isn't already being used to teach geometry, it ought to be.

posted by Rangi at 2:48 PM on June 20, 2014 [1 favorite]

posted by Rangi at 2:48 PM on June 20, 2014 [1 favorite]

I mean, there seem to be some buttons, representing drawing tools, at the top of the window. I try to use them to connect the two points, which seems to be the objective, but none of them seem to do that? What am I missing?

posted by JHarris at 3:35 PM on June 20, 2014

posted by JHarris at 3:35 PM on June 20, 2014

Wait, I guess the light-colored line is intended to be a "drawn" line, and not a tutorial indication of what needs to be done? So, I need to make a new point, and connect it to the others, to make a triangle, then?

posted by JHarris at 3:37 PM on June 20, 2014

posted by JHarris at 3:37 PM on June 20, 2014

Pro-Tip: always start by trying to construct something that

This is fantastic fun. Thanks boo_radley!

posted by straight at 3:44 PM on June 20, 2014

*looks*wrong. You have the solution when your process doesn't allow you to do it wrong.This is fantastic fun. Thanks boo_radley!

posted by straight at 3:44 PM on June 20, 2014

JHarris, yes, you need to make a new point, but you usually need to do some sort of construction first to locate the new point (by adding other objects, and placing the point at an intersection).

[spoilers] When I finally solved 20 it was by constructing the focus of the two circles and then the tangent through that focus, but I can't seem to get one based on this to work, and I really think it should. There are definitely a lot of solutions (with examples in the comments) that look right but are only approximate, but I don't think that's the case here.

posted by Pre-Taped Call In Show at 3:54 PM on June 20, 2014

[spoilers] When I finally solved 20 it was by constructing the focus of the two circles and then the tangent through that focus, but I can't seem to get one based on this to work, and I really think it should. There are definitely a lot of solutions (with examples in the comments) that look right but are only approximate, but I don't think that's the case here.

posted by Pre-Taped Call In Show at 3:54 PM on June 20, 2014

For instance, if you have to bisect a circle, start by drawing a line that obviously

posted by straight at 3:55 PM on June 20, 2014

*doesn't*bisect it.posted by straight at 3:55 PM on June 20, 2014

Once I figured out that the line on screen was one that existed, and not a guide, it fell into place quickly from there. Thanks though.

posted by JHarris at 3:56 PM on June 20, 2014

posted by JHarris at 3:56 PM on June 20, 2014

*When I finally solved 20 it was by constructing the focus of the two circles and then the tangent through that focus, but I can't seem to get one based on this to work, and I really think it should.*

Funnily enough, that looks like the solution I found. But it did require going back to make sure important points appeared in places that weren't too close to other points.

posted by eruonna at 4:02 PM on June 20, 2014

I must be missing something terribly simple on #14. I keep drawing the incircle, and it keeps not accepting it. I wish I could zoom in to see if I did it wrong somehow.

posted by mittens at 4:04 PM on June 20, 2014

posted by mittens at 4:04 PM on June 20, 2014

I'm stuck on level 11. But enjoying it. I haven't done geometry in over 30 years, and never this compass-and-straightedge stuff.

posted by not that girl at 5:05 PM on June 20, 2014

posted by not that girl at 5:05 PM on June 20, 2014

mittens, there is a very convincing almost-but not-quite-solution for 14, discussed in the comments there.

posted by Pre-Taped Call In Show at 5:10 PM on June 20, 2014

posted by Pre-Taped Call In Show at 5:10 PM on June 20, 2014

Argh. Stuck on #7. I'm fairly sure I have an answer (and drew the line segment), but it's not accepting it.

posted by hoyland at 5:22 PM on June 20, 2014

posted by hoyland at 5:22 PM on June 20, 2014

One problem (or, at least, potential source of confusion) with many of the levels is that the original objects are drawn in such a way that there are easy ways to make something thatLevel 20 has either a lot of almost-solutions that look okay but aren't quite on, or isn't very good at detecting alternate solutions. I thought I got it three times before I followed someone else's directions and finally officially got it.

*looks like*, but

*is not*, a solution. Level 20 is one such level. For example, try the following non-solution, and see what it looks like:

- Using the Compass tool, draw a circle of radius AB centered at point A.
- Using the Ray tool, draw a ray from point B through point A.
- Using the Intersection tool, determine the intersection between the circle you drew in (1) and the ray you drew in (2). For me, at least, this is labeled as point M.
- Again using the Intersection tool, determine the intersection between the circle you drew in (1) and the original circle centered at B. For me, at least, this is labeled as point N.
- Using the Segment tool, draw a line segment from M to N.

Well, it doesn't tell us that we got the solution because we

*didn't*get the solution.

To see this, clear the level and then do the exact same five steps over again, but this time, as you do each step, ask yourself the following question: "Does what I am doing now depend upon the radius of the original circle centered at point A?". You'll find that none of the steps depend on that original radius.

That is, even if the original circle centered at A was no bigger than a grain of sand, you would have drawn the exact same line segment MN as you would have drawn if the original circle centered at A was the size of Montana.

Clearly that one single line segment MN is not tangent to a circle centered at A the size of a grain of sand. And clearly it's not tangent to a circle centered at A the size of Montana. There is only one circle centered at A that it is tangent to, and the author of this puzzle happened to draw a circle that is very, very close to that size. If he had drawn a circle of basically any other size at all, you would not have been under the impression that the five steps I listed were an actual solution.

So, the moral of the story is: Just because it looks like you've got the answer doesn't mean you've got the answer.

There are actually several of these levels that have issues like this. I guess whether that's poor design or good design depends on your point of view; it can lead to frustration, but at the same time it can lead to thinking about your assumptions more deeply.

posted by Flunkie at 6:21 PM on June 20, 2014 [1 favorite]

Before you choose any of the tools, you can pick up points and move them around. I'm finding that starting with a different-looking triangle helps throw out some of those "looks right!" non-solutions.

posted by mittens at 6:25 PM on June 20, 2014

posted by mittens at 6:25 PM on June 20, 2014

This is perhaps due to another example of what I'm talking about: The dashed circle in your picture looks like it's the same size as the smaller of the two solid circles. But it's not (necessarily). In the real solution based on the idea in that picture, the radius of the dashed circle is the radius of the[spoilers] When I finally solved 20 it was by constructing the focus of the two circles and then the tangent through that focus, but I can't seem to get one based on this to work, and I really think it should.

*difference between*the radii of the two solid circles.

It happens to be drawn in that picture so that it looks to be the same size as the smaller of the two solid circles. So if you make a circle about point B with the same radius as the circle about point A, it's not going to tell you that you solved it. Because you didn't solve it - it only looks like you did due to the happenstance sizes of the original circles.

posted by Flunkie at 6:32 PM on June 20, 2014 [1 favorite]

>

You can do this even after you start, using the "Point on Object" tool. Click and drag on any of the original points, or any points that aren't completely constrained (which will be colored blue), and everything else adjusts based on how it was constructed.

posted by yuwtze at 7:29 PM on June 20, 2014 [3 favorites]

*Before you choose any of the tools, you can pick up points and move them around. I'm finding that starting with a different-looking triangle helps throw out some of those "looks right!" non-solutions.*You can do this even after you start, using the "Point on Object" tool. Click and drag on any of the original points, or any points that aren't completely constrained (which will be colored blue), and everything else adjusts based on how it was constructed.

posted by yuwtze at 7:29 PM on June 20, 2014 [3 favorites]

This is fantastic; thanks for sharing it! Finally beating level 20 was a joy!

I've taught constructions in Geometry and this would be a great help. However, the game doesn't allow an option that I've usually seen in pencil and paper constructions: when you've drawn a circle of some radius r and center P, you can then pick up the compass *while maintaining radius r* and draw another circle with radius r and center Q. This allows level 6 to be done like so: http://www.mathopenref.com/constparallel.html. I think I like the game's rules better.

posted by wiskunde at 7:32 PM on June 20, 2014

I've taught constructions in Geometry and this would be a great help. However, the game doesn't allow an option that I've usually seen in pencil and paper constructions: when you've drawn a circle of some radius r and center P, you can then pick up the compass *while maintaining radius r* and draw another circle with radius r and center Q. This allows level 6 to be done like so: http://www.mathopenref.com/constparallel.html. I think I like the game's rules better.

posted by wiskunde at 7:32 PM on June 20, 2014

*you would have drawn the exact same line segment MN as you would have drawn if the original circle centered at A was the size of Montana.*

This is exactly what I meant when I said try to draw something that looks wrong. Never try to eyeball a line or circle that looks about right. Draw something that will clearly only work if the geometric tools force it to work.

posted by straight at 7:37 PM on June 20, 2014

*when you've drawn a circle of some radius r and center P, you can then pick up the compass *while maintaining radius r* and draw another circle with radius r and center Q*

Seems like that would be okay as long as the student starts out by proving how to do the same thing exactly using a compass without that shortcut (it's essentially the same thing this game is doing).

posted by straight at 7:43 PM on June 20, 2014

Isn't that what Compass 2 does?

posted by Elementary Penguin at 8:06 PM on June 20, 2014

posted by Elementary Penguin at 8:06 PM on June 20, 2014

*Seems like that would be okay as long as the student starts out by proving how to do the same thing exactly using a compass without that shortcut (it's essentially the same thing this game is doing).*

Oh, yes, definitely. I believe the shortcut versions are the ones required on exams, in fact. I just enjoyed not having the shortcut because it forced me to think differently about the constructions.

posted by wiskunde at 8:11 PM on June 20, 2014

That's what the game's "Compass 2" tool is. It's only unlocked after level 8 though, which is where you prove you know how to do it ("Construct a circle with radius equal to line segment AB and center C").I've taught constructions in Geometry and this would be a great help. However, the game doesn't allow an option that I've usually seen in pencil and paper constructions: when you've drawn a circle of some radius r and center P, you can then pick up the compass *while maintaining radius r* and draw another circle with radius r and center Q.

The tool unfortunately doesn't work in all situations - I think something like if you define the radius by picking two points on a line segment that is parallel to a defined line segment on the new center that you pick, the game doesn't actually draw the new circle.

posted by Flunkie at 8:12 PM on June 20, 2014

I got pretty stuck on on 14, so if you're like me, a hint that would have helped is to remember that the circle is tangent to the sides where it touches. I had to look up a solution and only figured out why that worked after staring at the solution for a bit.

posted by vibratory manner of working at 8:19 PM on June 20, 2014

posted by vibratory manner of working at 8:19 PM on June 20, 2014

Ah, okay, thanks! I hadn't experimented with compass 2, since the original compass worked well enough. Nice, it just adds more to the game!

posted by wiskunde at 8:19 PM on June 20, 2014

posted by wiskunde at 8:19 PM on June 20, 2014

*I must be missing something terribly simple on #14. I keep drawing the incircle, and it keeps not accepting it. I wish I could zoom in to see if I did it wrong somehow.*

Wow, I just realized one of the coolest things you can do is use the "Point on Object" tool (the alternate function of the Intersect tool) and grab any point in your diagram and stretch the whole thing.

So for #14, you can draw the solution that looks like it might be right, but then if you grab the vertices of the the triangle and stretch it into a different triangle, you can see that your "solution" is obviously wrong for some triangles. You're looking for a general solution that works for any triangle.

That right there is an incredibly useful tool for getting a feel for what you're really trying to do with a geometric proof. It's not just solving a problem, it's solving every possible version of a problem.

posted by straight at 9:22 PM on June 20, 2014 [1 favorite]

hoyland: "

Took me a long time to figure this out (faintest of spoilers:) Draw your final segment in the other direction (IE: if you are clicking C then D instead click D then C). Not sure if this is a bug or a specific geometry jargon trip up.

posted by Mitheral at 10:38 PM on June 20, 2014

*Argh. Stuck on #7. I'm fairly sure I have an answer (and drew the line segment), but it's not accepting it.*"Took me a long time to figure this out (faintest of spoilers:) Draw your final segment in the other direction (IE: if you are clicking C then D instead click D then C). Not sure if this is a bug or a specific geometry jargon trip up.

posted by Mitheral at 10:38 PM on June 20, 2014

Flunkie:

*Groan* oh god of course. Thanks.

Resizing elements is incredibly useful... I wish I'd figured that out earlier

posted by Pre-Taped Call In Show at 1:45 AM on June 21, 2014

*the radius of the dashed circle is the radius of the difference between the radii of the two solid circles.**Groan* oh god of course. Thanks.

Resizing elements is incredibly useful... I wish I'd figured that out earlier

posted by Pre-Taped Call In Show at 1:45 AM on June 21, 2014

Why isn't my solution to #11 being recognized?

1. Circle2 AC to D. ("Draw a circle with radius AC and centered on point D.")

2. Draw a point J at the intersection between the circle and the ray from D.

3. Draw a line DJ. (Formalizing the line along the ray from D. The game says "well done" at this point.)

4. Circle2 CB to J. ("Draw a circle with radius CB and centered on point J.")

5. Draw a point K at the intersection between the circles centered on points D and J.

6. Draw a line DK.

This should solve it. The angle between DJ and DK is the same as between AB and AC. If I draw lines JK and BC, then DJK is congruent with ABC.

What am I missing?

posted by JHarris at 3:11 AM on June 21, 2014

1. Circle2 AC to D. ("Draw a circle with radius AC and centered on point D.")

2. Draw a point J at the intersection between the circle and the ray from D.

3. Draw a line DJ. (Formalizing the line along the ray from D. The game says "well done" at this point.)

4. Circle2 CB to J. ("Draw a circle with radius CB and centered on point J.")

5. Draw a point K at the intersection between the circles centered on points D and J.

6. Draw a line DK.

This should solve it. The angle between DJ and DK is the same as between AB and AC. If I draw lines JK and BC, then DJK is congruent with ABC.

What am I missing?

posted by JHarris at 3:11 AM on June 21, 2014

*The angle between DJ and DK is the same as between AB and AC. If I draw lines JK and BC, then DJK is congruent with ABC.*

It's not actually congruent. You're not guaranteed that the angle DJK is the same as the angle ACB (why, I can't articulate--I suck at this), so you don't actually have side-angle-side for congruent triangles, you've got to use one of those other congruence theorems.

posted by hoyland at 3:45 AM on June 21, 2014

JHarris: Your solution assumes that AB and AC are the same length.

posted by vibratory manner of working at 5:32 AM on June 21, 2014

posted by vibratory manner of working at 5:32 AM on June 21, 2014

Addictive! I would love to know how something like this would go over in an actual geometry class. "Unlocking a tool" is just so much more fun and interactive than "proving a theorem".

posted by Vitamaster at 6:53 AM on June 21, 2014 [1 favorite]

posted by Vitamaster at 6:53 AM on June 21, 2014 [1 favorite]

When I was in college I took an online self-paced java applet-driven predicate logic class. You started with three axioms and Modus Ponens, and spent the course working your way through all the usual theorems of predicate calculus. As you proved theorems (law of the excluded middle, etc) and new deductive methods (Modus Tolens, etc), you unlocked the ability to use them in subsequent problems. Also, you could work forward from the premise or backwards from the conclusion until your two lines of thought met up somewhere in the middle (hopefully).

It looks like this is the descendent of the class I took, if anyone is interested.

posted by Elementary Penguin at 8:00 AM on June 21, 2014 [3 favorites]

It looks like this is the descendent of the class I took, if anyone is interested.

posted by Elementary Penguin at 8:00 AM on June 21, 2014 [3 favorites]

JHarris, that's another example of the designer having drawn things in such a way that there are easy ways to make non-solutions that sure look like solutions. Try this:

Do your steps, exactly as you described. Then select the "point on object" tool, click on the point B, and drag it around. That will also cause the point K to move, since you defined it in terms of B.

If your answer is really a solution, as opposed to something that happens to look like a solution for this one specific case, then no matter where you drag B, K will move such that the two angles remain equal.

posted by Flunkie at 9:24 AM on June 21, 2014 [1 favorite]

Do your steps, exactly as you described. Then select the "point on object" tool, click on the point B, and drag it around. That will also cause the point K to move, since you defined it in terms of B.

If your answer is really a solution, as opposed to something that happens to look like a solution for this one specific case, then no matter where you drag B, K will move such that the two angles remain equal.

posted by Flunkie at 9:24 AM on June 21, 2014 [1 favorite]

This is a lot of fun and my household mathematician is now thinking about ways to include it in the curriculum for the math major. Thanks for posting it!

posted by LobsterMitten at 11:28 AM on June 21, 2014

posted by LobsterMitten at 11:28 AM on June 21, 2014

Oh my goodness, the sense of accomplishment after finally completing #20! It took HOURS, but how satisfying.

posted by mittens at 11:59 AM on June 21, 2014

posted by mittens at 11:59 AM on June 21, 2014

Great game, really brought back some fond memories of grade school geometry.

Really wish I knew about the "click and drag points" thing before I reached level 20 though. It's really nifty, and it's helpful to show off how your almost-solutions are actually frauds.

posted by cyberscythe at 12:14 PM on June 21, 2014 [2 favorites]

Really wish I knew about the "click and drag points" thing before I reached level 20 though. It's really nifty, and it's helpful to show off how your almost-solutions are actually frauds.

posted by cyberscythe at 12:14 PM on June 21, 2014 [2 favorites]

Really great game. I "found" a solution for level 20, but had absolutely no idea why it worked. Spent about an hour thinking about it, and had one of the most satisfying AHA! moments of my life once I actually understood it. Excellent game. Really hope more levels are made.

posted by cthuljew at 3:07 PM on June 21, 2014

posted by cthuljew at 3:07 PM on June 21, 2014

Oh, and he is apparently updating it...I went back in this morning and found a slightly different button layout, and some improved graphics.

posted by mittens at 8:54 AM on June 22, 2014

posted by mittens at 8:54 AM on June 22, 2014

I'm finding that I wish there were an easy way to review my solutions to previous levels, so I can remember the whole "proof"/construction from the ground up for some of these later more complicated levels.

posted by LobsterMitten at 10:57 AM on June 22, 2014

posted by LobsterMitten at 10:57 AM on June 22, 2014

Would someone mind explaining why the following construction doesn't work for #20?

You have circles centered at A and B. First, construct a new circle centered at B that has the same radius as the circle centered at A. Call this the

Now, construct tangents to the inner circle centered at B that go through point A as follows. First, take the midpoint M between A and B. Then draw a circle centered at M and having radius of length(A, M). Then draw points at the intersections between the circle centered at M and the inner circle centered at B. Call those points P and Q. Finally, draw a segment from P to A and a segment from Q to A. Those should (I think) be tangents to the inner circle centered at B that go through A, by Thales' Theorem.

Now, draw lines through A that are perpendicular to those tangents you just found. Take the points where those perpendiculars intersect with the circle centered at A. Similarly, draw lines through B that are perpendicular to the tangents found above, and take the points where those perpendiculars intersect with the outer circle centered at B. Join the points in the obvious ways.

That should give the outer tangents ... shouldn't it?

When I carry out these operations, the diagram doesn't look right, but the proofs

posted by Jonathan Livengood at 8:20 PM on June 22, 2014

You have circles centered at A and B. First, construct a new circle centered at B that has the same radius as the circle centered at A. Call this the

*inner circle*centered at B, and call the original circle centered at B the*outer circle*centered at B.Now, construct tangents to the inner circle centered at B that go through point A as follows. First, take the midpoint M between A and B. Then draw a circle centered at M and having radius of length(A, M). Then draw points at the intersections between the circle centered at M and the inner circle centered at B. Call those points P and Q. Finally, draw a segment from P to A and a segment from Q to A. Those should (I think) be tangents to the inner circle centered at B that go through A, by Thales' Theorem.

Now, draw lines through A that are perpendicular to those tangents you just found. Take the points where those perpendiculars intersect with the circle centered at A. Similarly, draw lines through B that are perpendicular to the tangents found above, and take the points where those perpendiculars intersect with the outer circle centered at B. Join the points in the obvious ways.

That should give the outer tangents ... shouldn't it?

When I carry out these operations, the diagram doesn't look right, but the proofs

*do*look right. So, I'm not sure if I'm just screwing something up in the proofs or if the diagram / program is screwy or both.posted by Jonathan Livengood at 8:20 PM on June 22, 2014

>

Flunkie describes it above, but as I understand it, the inner circle needs to be of radius r

posted by yuwtze at 9:12 PM on June 22, 2014

*Would someone mind explaining why the following construction doesn't work for #20*Flunkie describes it above, but as I understand it, the inner circle needs to be of radius r

_{b}-r_{a}, not r_{a}. The construction you describe works only for the special case r_{b}= 2r_{a}. The game designer has deviously set up the initial points in exactly that special case, but if you drag the points around (right-click-and-drag), you'll quickly see that your solution doesn't work in the general case.posted by yuwtze at 9:12 PM on June 22, 2014

Yeah, I should have been clearer. I get what you and Flunkie are saying, but it's not helping me understand why the inner circle radius needs to be the difference between rb and ra, rather than just ra. Put another way: could you explain why a translation of tangents works in the one case but not in the other?

posted by Jonathan Livengood at 10:53 PM on June 22, 2014

posted by Jonathan Livengood at 10:53 PM on June 22, 2014

Jonathan Livengood, your solution takes a tangent on a circle of size r

posted by Joe in Australia at 5:04 AM on June 23, 2014

_{a}and displaces is by r_{a}, so now it's tangent to a non-existent circle of size 2 r_{a}. If your inner circle was instead r_{b}- r_{a}then the two r_{a}terms would cancel out when you displace it by r_{a}, leaving you with what you want: r_{b}. As yuwtze says, it was quite devious to make the bigger circle appear to be almost 2 r_{a}, because it made your answer look almost correct.posted by Joe in Australia at 5:04 AM on June 23, 2014

Another way of saying the same thing in a way that may or may not be clearer:

You've created a line segment that's tangent to the inner circle. You're then grabbing that line segment by both ends, and moving both of those ends in a direction perpendicular to that tangent. But you're moving one of them one distance in that direction, and the other an entirely different distance in that same direction. So you've not only moved the line segment - you've also rotated it (and lengthened or compressed it). Therefore, after you've finished moving that segment, it is no longer parallel to the original tangent.

And if the line segment is no longer parallel to the original tangent, then it's not perpendicular to the radius of the inner circle by which that was defined.

And the line containing that radius of the inner circle is the same line that contains the radius of the outer circle intersecting the new location of the line segment. So the segment is not perpendicular to that radius of that outer circle, either.

And if a line intersects a circle at some point, but is not perpendicular to the circle's radius that intersects that point, then the line is not tangent to the circle.

posted by Flunkie at 11:20 AM on June 23, 2014

You've created a line segment that's tangent to the inner circle. You're then grabbing that line segment by both ends, and moving both of those ends in a direction perpendicular to that tangent. But you're moving one of them one distance in that direction, and the other an entirely different distance in that same direction. So you've not only moved the line segment - you've also rotated it (and lengthened or compressed it). Therefore, after you've finished moving that segment, it is no longer parallel to the original tangent.

And if the line segment is no longer parallel to the original tangent, then it's not perpendicular to the radius of the inner circle by which that was defined.

And the line containing that radius of the inner circle is the same line that contains the radius of the outer circle intersecting the new location of the line segment. So the segment is not perpendicular to that radius of that outer circle, either.

And if a line intersects a circle at some point, but is not perpendicular to the circle's radius that intersects that point, then the line is not tangent to the circle.

posted by Flunkie at 11:20 AM on June 23, 2014

Got it. Thanks guys.

posted by Jonathan Livengood at 11:26 AM on June 24, 2014

posted by Jonathan Livengood at 11:26 AM on June 24, 2014

« Older Gilbert Legrand | The Fansubbed Last Words of an Auditory Phantom Newer »

This thread has been archived and is closed to new comments

this is amazingI am even wearing my geometry t-shirt

posted by Pre-Taped Call In Show at 9:59 AM on June 20, 2014 [1 favorite]