MathDB

Concurrency

Source: Brazil MO 2018 Grades 8 and 9

November 16, 2018

geometrycircumcircle

Problem Statement

Let

ABC

be an acute-angled triangle with circumcenter

O

and orthocenter

H

. The circle with center

X_a

passes in the points

A

and

H

and is tangent to the circumcircle of

ABC

. Define

X_b, X_c

analogously, let

O_a, O_b, O_c

the symmetric of

O

to the sides

BC, AC

and

AB

, respectively. Prove that the lines

O_aX_a, O_bX_b, O_cX_c

are concurrents.

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