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Two points lying on a triangle's circumcircle

Source: 2023 Centroamerican and Caribbean Math Olympiad, P5

July 26, 2023

geometrycircumcircle

Problem Statement

Let

ABC

be an acute-angled triangle with

AB < AC

and

\Gamma

the circumference that passes through

A,\ B

and

C

. Let

D

be the point diametrically opposite

A

on

\Gamma

and

\ell

the tangent through

D

to

\Gamma

. Let

P, Q

and

Q D

be the intersection points of

B C

with

\ell

, of

A P

with

\Gamma

such that

Q \neq A

and of

A C

with the

\ell

-altitude of the triangle

S

, respectively. Define

S

to be the intersection of

AB

with

\ell

and

T

to be the intersection of

A C

with

\ell

. Show that

S

and

T

lie on the circumference that passes through

A, Q

and

R

.

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