Hints and Solutions: Patty Paper Trisection

No peeking! This post is for those of you who have given the trisection proof a good workout on your own. If you have a question about the proof or a solution you would like to share, please post a comment here.

But if you haven’t yet worked at the puzzle, go back and give it a try. When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.

The Bestselling Math Book of All Time


When all else fails, ask a teacher — and for proofs, who better than Euclid? His geometry textbook, the Elements, is the bestselling math textbook of all time. Here are some tips from the master:

Book I, Common Notions
Things which equal the same thing also equal one another.

Book 1, Proposition 5
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

Book 1, Proposition 29
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.

Book 1, Proposition 32
In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.

Book 1, Proposition 33
Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

That “Aha!” Feeling

Hint for trisection proof

When I tried this proof for myself, I thought it was relatively easy to prove 2 of the small angles congruent. But to show that they were both equal to the third angle was tough — and unless I could show that all 3 of the small angles were the same, my proof wasn’t worth anything. As I fidgeted with my patty paper, I noticed that two lines seemed to overlap.

(Click the image for a larger view.)

My geometry teacher in high school drilled one rule repeatedly, until it was permanently embedded in my brain: Never trust the drawing! But what if it wasn’t just a coincidence? What if the lines really did match?

Aha! That was my key. As soon as I could show that those two lines would always coincide, no matter what the original angle was, the rest of my proof was a cinch.

[Hat tip: Trisecting an Angle with Origami at Math Trek, via the Carnival of Mathematics.]

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