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Orthocentre,mixed with median and symmedian

Source: STEMS 2024 CAT A P5,CAT B P4

December 17, 2023

geometry

Problem Statement

Let ABC with orthocenter

(ABC)

and circumcenter

O

be an acute scalene triangle satisfying

AB = AM

where

M

is the midpoint of

BC

. Suppose

Q

and

K

are points on

(ABC)

distinct from A satisfying

\angle AQH = 90

and

\angle BAK = \angle CAM

. Let

N

be the midpoint of

AH

. • Let

I

be the intersection of

B\text{-midline}

and

A\text{-altitude}

Prove that

IN = IO

. • Prove that there is point

P

on the symmedian lying on circle with center

B

and radius

BM

such that

(APN)

is tangent to

AB

.
Proposed by Krutarth Shah

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