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concyclic wanted, projections of interior point on sides of ABC related

Source: Switzerland - Swiss MO 2012 p10

July 16, 2020

geometryConcyclicprojectionsprojection

Problem Statement

Let

O

be an inner point of an acute-angled triangle

ABC

. Let

A_1, B_1

and

C_1

be the projections of

O

on the sides

BC, AC

and

AB

respectively . Let

P

be the intersection of the perpendiculars on

B_1C_1

and

A_1C_1

from points

A

and

B

respectilvey. Let

H

be the projection of

P

AB

. Show that points

A_1, B_1, C_1

and

H

lie on a circle.