MathDB

fixed point and perpendicular wanted, 2 circles, tangents

Source: 2017 Thailand October Camp 2.3

October 15, 2020

geometryfixedperpendicularFixed point

Problem Statement

Let

C

be a chord not passing through the center of a circle

\omega

. Point

A

varies on the major arc

BC

. Let

E

and

F

be the projection of

B

onto

AC

, and of

C

onto

AB

respectively. The tangents to the circumcircle of

\vartriangle AEF

at

E, F

intersect at

P

. (a) Prove that

P

is independent of the choice of

A

. (b) Let

H

be the orthocenter of

\vartriangle ABC

, and let

T

be the intersection of

EF

and

BC

. Prove that

TH \perp AP

.

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