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.
If ab
So AB=AC
Baidu encyclopedia-Steiner theorem