# What is the “Donkey Theorem”?

I was watching the Turkish version of Who Wants to Be a Millionaire? and they asked this question:

What field is the Donkey Case (or I guess it can be translated as Donkey Theorem) related to?

The correct answer was Mathematics. I've never heard of it. What is it?

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The very first Google hit for "Donkey theorem" is regentsprep.org/Regents/math/geometry/GP4/Ltriangles.htm . It apparently refers to angle-side-side (ASS) congruence. – Qiaochu Yuan Sep 20 '11 at 21:11
LOL! + lorem ipsum, lorem ipsum (to meet the minimum character count requirement - bah!) – Mike Jones Sep 20 '11 at 21:22
So long ago, I had forgotten. Ass is another word in English for Donkey, so Qiaochu has it. So, in church, they would say "riding on an ass," but the geometry teacher tried to avoid saying Ass because it also refers to the human posterior(in England probably Arse). Also it is not a theorem, it is false, so all comes full circle. Lord, I'm old. – Will Jagy Sep 20 '11 at 21:39
@Qiaochu Regardless of the first Google hit for Donkey Theorem, I do not believe that the ASS theorem is the right reference. The pons asinorum theorem of Euclid that an isoceles triangle has two equal angles (en.wikipedia.org/wiki/Pons_asinorum) is one possibility; another is the triangle inequality in Euclid's Book I. Some of Euclid's contemporaries claimed that the triangle inequality did not require a proof because "even a donkey knows that the shortest path from A to B does not pass through a (non-collinear) point C" – Dilip Sarwate Sep 20 '11 at 22:51
Dilip, it is not so much about actual ancient history as it is about what the people providing questions for a modern popular television show might be thinking. The fact that Mehper suggests "Case" as the most natural translation really seems to support the idea of a special case that people sometimes, foolishly, believe can still be handled in the same manner. So having this be the donkey case in congruent triangles is very attractive. – Will Jagy Sep 24 '11 at 1:41

The donkey theorem is also known as the triangle inequality theorem. It states that in a triangle ABC: a < b + c.

The name comes from the idea that if you have a donkey standing at side A, and a hay stack at side C, it will ALWAYS be a shorter path for the donkey to go straight from A to C instead of from A to B to C.

This theorem comes in handy when you're trying to find if it's possible to have a triangle with side lengths 44, 32, and 19, for example.

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Welcome to Math.SE! I'm afraid your proposed Answer to this older question is not very well written. By A,B,C do you mean the vertices or the sides of the triangle? It would make more sense for these to be vertices ("if you have a donkey standing at vertex A"), and for the lowercase a,b,c to denote the side lengths. – hardmath Apr 7 '13 at 1:33

Alternate interpretation of the case (historical novelty): There is no loop path involving a specific set of 3 land masses and five bridges crossing each bridge only once. (I don't have the link but the original question included a map of a real city on an island).

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The Donkey Theorem is a humorous name for the theorem that triangles can't be proven congruent through ASS (angle, side, side). SSS, ASA, SAS, and AAS can be used to prove congruency because they are static shapes. There are two possible triangles to be made with angle, side, side congruency as shown by the swinging door. / \ /| /___\ = /_|

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