Problem

Source: China tst 2003

Tags: geometry, circumcircle, symmetry, projective geometry, power of a point, radical axis, geometry proposed



Denote by (ABC) the circumcircle of a triangle ABC. Let ABC be an isosceles right-angled triangle with AB=AC=1 and CAB=90. Let D be the midpoint of the side BC, and let E and F be two points on the side BC. Let M be the point of intersection of the circles (ADE) and (ABF) (apart from A). Let N be the point of intersection of the line AF and the circle (ACE) (apart from A). Let P be the point of intersection of the line AD and the circle (AMN). Find the length of AP.