Geometric theorem of Steiner theorem

Steiner Theorem: "If the bisectors of two internal angles in a triangle are equal in length, then it must be an isosceles triangle". The inverse proposition of this proposition: "The bisectors of the two base angles of an isosceles triangle are equal in length" has been regarded as a theorem in the Elements of Geometry more than 2000 years ago, which is easy to prove.

However, the above proposition was not mentioned in the Elements of Geometry until 1840, when Lemos asked to prove pure geometry in his letter to storm. Storm didn't solve it, so he asked many mathematicians this question. Swiss geometer J Steiner (1796 ~ 1863) first gave the proof, so this theorem is called Steiner-Lemos theorem.

Extended data:

Prove:

In △ABC, BD and CE are angular bisectors, and BD=CE.

Let ∠ Abd = ∠ CBD = X, ∠ ACE = ∠ BCE = Y.

According to spread angle theorem, there is

2cosx/BD= 1/AB+ 1/BC

2 osy/CE = 1/AC+ 1/BC

Then 2 * AB * BC * COSX/(AB+BC) = BD = CE = 2 * AC * BC * COSY/(AC+BC)

That is, (ab * (AC+BC))/(AC * (AB+BC)) = cosy/cosx.

Using fractional theorem. Sum-difference product is used for cosy-cosx.

a b-AC =(-(2 * AC *(a b+BC))/(BC * cosx))* sin((y+x)/2)* sin((y-x)/2)

If AB & gtAC, the left end of the above formula is positive and the right end is negative.