377. Some remarkable properties of the symmedian.
by Virgil Nicula, Jun 8, 2013, 4:44 PM
PP1. Let 
be a triangle with the circumcircle 
. Denote 
- the tangent line to the circle 
at the point 
. The 
-symmedian cut 
in 
and
meet again in 
. Denote the midpoint 
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, 
, 
and 
, 
. Prove that the following relations:
. Generally, if 
is the -isogonal line of the line 
for which 
, then 
.
and 
.
(the quadrilateral 
is harmonically) and 
.
. Particular case. If 
, then 
.
Proof.
Is well-known that 
, i.e. 
. Denote 
.
Apply the theorem of Sines in the triangles:
.
and 
, i.e. the quadrilateral is harmonically: 
.
Observe that
and 
, i.e. 
.
Denote
. Prove similarly 
.
Denote
. Therefore, 
.
Apply the remarkable inequality 
to 
. We have the equality if and only if
the triangle
is equilateral 
and 
and 
.
be a triangle with the circumcircle 
. Denote 
- the tangent line to the circle 
at the point 
. The 
-symmedian cut 
in 
andmeet again
in 
. Denote the midpoint 
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, 
, 
and 
, 
. Prove that the following relations:
. Generally, if 
is the -isogonal line of the line 
for which 
, then 
.
and 
.
(the quadrilateral 
is harmonically) and 
.
. Particular case. If 
, then 
.Proof.
Is well-known that 
, i.e. 
. Denote 
.Apply the theorem of Sines in the triangles:
.
and 
, i.e. the quadrilateral is harmonically: 
.Observe that
and 
, i.e. 
.Denote
. Prove similarly 
.Denote
. Therefore, 
.
Apply the remarkable inequality 
to 
. We have the equality if and only ifthe triangle
is equilateral 
and 
and 
.This post has been edited 31 times. Last edited by Virgil Nicula, Nov 15, 2015, 1:51 PM
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