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Fixed Circumcenter

Source: Problem 1, China National High School Mathematics League 2023

September 10, 2023

geometrycircumcirclechang09

Problem Statement

Let

A,B

be two fixed points on a plane and

\Omega

a fixed semicircle arc with diameter

AB

. Let

T

be another fixed point on

\Omega

, and

\omega

a fixed circle that passes through

C

and

AP

and has its center in

\Delta ABT

. Let

P

be a moving point on the arc

TB

(endpoints excluded), and

C,D

be two moving points on

\omega

such that

C

lies on segment

AP

,

C,D

lies on different sides of line

AB

and

CD\ \bot \ AB

. Denote the circumcenter of

\Delta CDP

of

K

. Prove that (i)

K

lies on the circumcircle of

\Delta TDP

. (ii)

K

is a fixed point.

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