Question 73
Triangles $PAB$ and $PCD$ are placed on a plane. Suppose that $PA = PB$, $PC = PD$ are satisfied and that the 3 points $P$, $A$, $C$ lie on a straight line in this order and the same is true for the 3 points $B$, $P$, $D$. Suppose further that a circle $S_1$ going through $A$, $C$ and the circle $S_2$ going through $B$, $D$ intersect at 2 distinct points $X$ and $Y$. Prove that the orthocenter of the triangle $PXY$ coincides with the mid-point of the line segment connecting the centers of the circles $S_1$ and $S_2$.
Here, we denote for a line segment $ZW$ its length also by $ZW$.
Here, we denote for a line segment $ZW$ its length also by $ZW$.