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Arc midpoints in the right triangle

Source: Kyiv City MO 2023 Round 1, Problem 9.3

December 16, 2023

geometrycircumcircle

Problem Statement

You are given a right triangle

ABC

with

\angle ACB = 90^\circ

. Let

W_A , W_B

respectively be the midpoints of the smaller arcs

BC

and

AC

of the circumcircle of

\triangle ABC

, and

N_A , N_B

respectively be the midpoints of the larger arcs

BC

and

AC

of this circle. Denote by

P

and

Q

the points of intersection of segment

AB

with the lines

N_AW_B

and

N_BW_A

, respectively. Prove that

AP = BQ

.
Proposed by Oleksiy Masalitin

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