MathDB

Find AP [in an isosceles right triangle ABC]

Source: China tst 2003

June 8, 2005

geometrycircumcirclesymmetryprojective geometrypower of a pointradical axisgeometry proposed

Problem Statement

Denote by

\left(ABC\right)

the circumcircle of a triangle

ABC

. Let

ABC

be an isosceles right-angled triangle with

AB=AC=1

and

\measuredangle CAB=90^{\circ}

. 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

\left(ADE\right)

and

\left(ABF\right)

(apart from

A

). Let

N

be the point of intersection of the line

AF

and the circle

\left(ACE\right)

(apart from

A

). Let

P

be the point of intersection of the line

AD

and the circle

\left(AMN\right)

. Find the length of

AP

.

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