Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Compare Areas

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

Bi-cyclics

Stage: 4 Challenge Level:

Two circles intersect at $A$ and $B$. $C$ and $D$ are points on one circle and they can be moved around the circle. The line $CA$ meets the second circle in $E$. The line $DB$ meets the second circle in $F$.

As $C$ and $D$ move around one circle what do you notice about the line segments $CD$ and $EF$?