Dr. Gene Chase guest blog author here again.
What makes a math theorem important?
The usual answer is that it is either beautiful or useful. If like me you think that being useful is a beautiful thing, then important theorems are the beautiful ones.
But what makes a theorem beautiful? For example, why is the Theorem of Pythagoras widely regarded as beautiful: 
But the theorem 1223334444 = 1223334443 + 1 is not beautiful, won’t you agree?
If the theorem is geometric, we can appeal to visual beauty. For example, three circles pairwise tangent have a beautiful property that is animated here.
But beautiful theorems do not have to be geometric. Numbers are beautiful. For example, Euclid’s theorem that there are an infinity of primes is beautiful. No one has been able to draw a beautiful picture about that, although people have tried from astronomer and mathematician Eratosthenes in 200 BC to science fiction writer and mathematician Stanislaw Ulam in 1963.
For $15 you can have a mathematical theorem named after you. But I can guarantee that it won’t be beautiful. So if you want a theorem named after you, give Mr. Chase the $15 instead and he’ll find one for you. Don’t use 1223334444 = 1223334443 + 1. I claim that as “Dr. Gene Chase’s theorem.”
Answer to discussion question above: Most folks say that a beautiful theorem has to be “deep,” which is just a metaphor for “having many connections to many other things.” For example, the Theorem of Pythagoras has to do with areas, not squares specifically. The semicircle on the hypotenuse of a right triangle has an area equal to the sum of the areas of the semicircles on the adjacent sides. And so for any three similar figures.
Do you remember the joy that you feel when you first learned that two of your friends are also friends of each other? That’s the joy that a mathematician feels when she discovers that the Theorem of Pythagoras and the Theorem of Euclid are intimate with each other. But I’ll leave that connection to another post.
Math is about surprising connections. Which is to say, it’s about beauty.
Random Walks 
You just stopped short from beautifully proving the Pythagorean theorem on the basis of your remark. This would have been Euclid VI.31, see http://www.cut-the-knot.org/pythagoras/euclid.shtml
On the sides of a right triangle form triangles similar to the given one but turned inside. You get this consruction by dropping the altitude from the right angle. Now because two smaller triangles snugly fit into the larger one, the construction proves the theorem.
Honored to have the author of Cut-The-Knot.org stop by!
Euclid VI.31 was in my first draft, but hit the cutting room floor, as did the thing about connecting primes and Pythagoras that I mentioned I hope to discuss later.
Thank you for the kind reply. And sorry if I have disturbed your plans. Euclid VI.31 is my tender spot.
It’s always a pleasure to read your blog, not to mention the wonderful father-and-son cooperation.
Yes, I’m a huge fan of your site too, Alexander.
I’m currently advising a student on his capstone paper (for IB), and he chose the Pythagorean Theorem as his topic. He provides multiple proofs, including the internal-triangles proof. So I’ve thought about this proof as recently as last month. A very elegant proof indeed!
Pingback: Pythagorean Theorem « Random Walks
Pingback: The Important Theorems Are the Beautiful Ones | Random Walks | mathbeauty