be integer numbers satisfy that: 
.
and 
be isogonal conjugates with respect to 
.
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 
.
?
of an isosceles right-angled triangle 
such 
and 
are marked, such that 
.
.
Y by Adventure10
A circle 
is given with a non-diameter chord 
. On the tangent at point 
select point 
and mark 
the intersection of the circle with the interior of the line 
(if any). Let 
a point in circle such that quadrilateral 
is a trapezoid . Determine the set of intersections of lines 
and 
belonging to all such trapezoids.
is given with a non-diameter chord 
. On the tangent at point 
select point 
and mark 
the intersection of the circle with the interior of the line 
(if any). Let 
a point in circle such that quadrilateral 
is a trapezoid . Determine the set of intersections of lines 
and 
belonging to all such trapezoids.This post has been edited 1 time. Last edited by parmenides51, Jan 31, 2020, 11:24 AM
