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perpendicular bisectors featuring circles

Source: Sharygin 2023 - P12 (Grade-(a)8-9, (b)10-11)

March 4, 2023

geometryperpendicular bisectororthologic trianglesSharygin Geometry OlympiadSharygin 2023

Problem Statement

Let

ABC

be a triangle with obtuse angle

B

, and

P, Q

lie on

AC

in such a way that

AP = PB, BQ = QC

. The circle

BPQ

meets the sides

AB

and

BC

at points

N

and

M

respectively.

\qquad<span class='latex-bold'>(a)</span>

(grades 8-9) Prove that the distances from the common point

R

of

PM

and

NQ

to

A

and

C

are equal.

\qquad<span class='latex-bold'>(b)</span>

(grades 10-11) Let

BR

meet

AC

at point

S

. Prove that

MN \perp OS

, where

O

is the circumcenter of

ABC

.

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