be a triangle in which 
.
and 
be the incentre and circumcentre of 
of the circumcircle of 
.
given that 
.
.
and 
intersect at point 
is symmetric to point 
intersects the circumcircle 
of the triangle 
at the point 
.
is tangent to the circle 
.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
and 
intersect at the point 
and 
.
intersect at point 
be the reflection of 
and 
intersect at point 
.
.
Y by Adventure10, Mango247
The inscribed circle of an acute 
is tangent to 
and 
in 
and 
respectively. The altitude 
intersects the angle bisectors of 
and 
in 
and 
respectively. Prove that the middle point 
of lies on the radical axis of the circumscribed circles of 
and 
.
is tangent to 
and 
in 
and 
respectively. The altitude 
intersects the angle bisectors of 
and 
in 
and 
respectively. Prove that the middle point 
of lies on the radical axis of the circumscribed circles of 
and 
.
