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November 10th, 2017, 08:53 AM   #1
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angle bisectors -open problem?

Equality of angles' bisectors' lengths in a triangle leads to an isosceles triangle is a well-known theorem.
The only proofs are based on using the lengths formula of angle bisectors or proving by contradiction. A direct proof, surprisingly, is still challenging!
Is a direct way to prove it (without the lengths formula) still unknown?

Last edited by skipjack; November 10th, 2017 at 06:36 PM.
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November 10th, 2017, 07:14 PM   #2
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The theorem is called the Steiner–Lehmus theorem. It seems somewhat artificial to require a proof that doesn't use a particular formula. Also, what exactly do you mean by a "direct" proof, given that various "standard" theorems of geometry that one might like to use in a proof are usually proved "indirectly" by contradiction?

There's a geometrical proof given in this article. Do you consider it to be "direct"?

Last edited by skipjack; November 10th, 2017 at 08:00 PM.
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