November 10th, 2017, 07:53 AM  #1 
Newbie Joined: Nov 2017 From: Israel Posts: 1 Thanks: 0  angle bisectors open problem?
Equality of angles' bisectors' lengths in a triangle leads to an isosceles triangle is a wellknown 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 05:36 PM. 
November 10th, 2017, 06:14 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,708 Thanks: 1805 
The theorem is called the Steinerâ€“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 07:00 PM. 

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angle, bisectors, open, problem 
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