37. Own extension of the Steiner-Lehmus' problem.
by Virgil Nicula, May 8, 2010, 6:49 PM
Virgil Nicula wrote:
Extension of Steiner-Lehmus' problem. Let 
and let 
for which the ray 
is the bisector of the angle
. For 
denote 
and 
. Prove that 
.
and let 
for which the ray 
is the bisector of the angle
. For 
denote 
and 
. Prove that 
.Proposed problem. Acute angled triangle
is given. A line 
parallel to side 
passing through vertex 
is drawn. Let the angle bisectors of 
and
intersect the sides 
and 
at points 
and 
and line at points 
and 
respectively. Prove that if 
then 
.Proof.Define the following relation between any two real numbers
, 
(same sign) .Denote the length
of the 
-bisector a.s.o. Prove easily that 
and 
.In conclusion,
and 
=====================================================================================
Indeed, 
a.s.o. and 
![$a(s-b)(b+c)^2=(b-a)\left[s(c^2-ab)+ab(a+b+2c)\right]\equiv (b-a)\cdot E$](http://latex.artofproblemsolving.com/d/2/1/d21220e76417e41a567929cb5a89525b359359a0.png)
. Observe that 
because
.Proposed problem. Let
be a triangle and let be the intersection of -symmedian with the sideline .For a point
denote 
, 
. Prove that 
.Remark. In particular case when
- the symmedian point of obtain a similar result with Steiner-Lehmus' theorem for the incenter 
.This post has been edited 9 times. Last edited by Virgil Nicula, Nov 23, 2015, 8:43 AM
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