AoPS Academy
be a convex quadrilateral. The perpendicular bisectors of its sides 
and 
meet at 
. Denote by 
a point inside the quadrilateral such that 
and 
. Show that 
.
such that
for all nonnegative real numbers 
, 
, and 
with 
.
be any point in the plane of triangle 
. Let 
be the inradius of , let 
, 
, 
be the lengths of the altitudes from 
, 
, 
, respectively, and let 
for 
. Prove 
.
be the circumcenter, symmedian point of 
. Show that the tangent of 
at ,the tangent of 
at , the tangent of 
at are concurrent.
, you can change symmedian point to any point on Jerabek hyperbola.
with circumcenter 
and a point 
lying on the Jerabek hyperbola of 
Let 
be the tangent of 
at 
respectively. Then are concurrent.
be the antigonal conjugate of WRT and 
be the image of under the inversion WRT 
then from 
we conclude that lies on 
