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CA, AP and PE are the side lengths of a right triangle

Source: 2013 China Mathematical Olympaid P1

January 12, 2013

geometrycircumcircleperpendicular bisectorpower of a pointgeometry proposed

Problem Statement

Two circles

K_1

and

K_2

of different radii intersect at two points

A

and

B

, let

DB

and

D

be two points on

K_1

and

K_2

, respectively, such that

A

is the midpoint of the segment

CD

. The extension of

DB

meets

K_1

at another point

E

, the extension of

CB

meets

K_2

at another point

F

. Let

l_1

and

l_2

be the perpendicular bisectors of

CD

and

EF

, respectively. i) Show that

l_1

and

l_2

have a unique common point (denoted by

P

). ii) Prove that the lengths of

CA

,

AP

and

PE

are the side lengths of a right triangle.

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