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Obtuse angle at variable midpoint

Source: Sharygin Finals 2017, Problem 9.3

August 3, 2017

geometric inequalitygeometry

Problem Statement

The angles

B

and

A,P,Q

of an acute-angled triangle

ABC

are greater than

60^\circ

. Points

P,Q

are chosen on the sides

AB,AC

respectively so that the points

A,P,Q

are concyclic with the orthocenter

H

of the triangle

ABC

. Point

K

is the midpoint of

PQ

. Prove that

\angle BKC > 90^\circ

.
Proposed by A. Mudgal

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