The Steiner-Lehmus Theorem states Every triangle with two angle bisectors of equal lengths is isosceles.
Many proofs end with showing the base angles of the triangle are equal. Equality may be asserted by shows the sines, cosines, or other trigonometric functions are equal thus the angles are equal (within proscribed ranges).
The ultimate (or assumed) appeal is to Euclid 1.6. "If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another."
But Euclid 1.6 is the first proof by contradiction. Thus any use of 1.6 is an indirect proof.
Euclid 1.6 has an wide-ranging effect. With the help of the Resource "Theorem Network from Euclid's Elements" one finds there are 63 propositions in Euclid that depend on 1.6. The challenge for a direct proof is to find a relationship between sides on a triangle not dependent on angle.
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