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14th ibmo - cuba 1999/q5.

Source: Spanish Communities

April 16, 2006

geometrycircumcirclesymmetrygeometry unsolved

Problem Statement

An acute triangle

\triangle{ABC}

is inscribed in a circle with centre

O

. The altitudes of the triangle are

AD,BE

and

CF

. The line

EF

cut the circumference on

P

and

Q

. a) Show that

OA

is perpendicular to

PQ

. b) If

M

is the midpoint of

BC

, show that

AP^2=2AD\cdot{OM}

.

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