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Strange angle equality

Source: Russian TST 2015, Day 8 P2 (Group NG), P3 (Groups A & B)

April 21, 2023

geometryangles

Problem Statement

The triangle

ABC

is given. Let

P_1

and

P_2

be points on the side

AB

such that

P_2

lies on the segment

BP_1

and

AP_1 = BP_2

. Similarly,

Q_1

and

Q_2

are points on the side

BC

such that

Q_2

lies on the segment

BQ_1

and

BQ_1 = CQ_2

. The segments

P_1Q_2

and

P_2Q_1

intersect at the point

R{}

, and the circles

P_1P_2R

and

Q_1Q_2R

intersect a second time at the point

S{}

lying inside the triangle

P_1Q_1R

. Let

M{}

be the midpoint of the segment

AC

. Prove that the angles

P_1RS

and

Q_1RM

are equal.

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