MathDB

Orthocenter is the midpoint of the altitude

Source: Mexican Quarantine Mathematical Olympiad P4

April 26, 2020

geometryaltitudeorthocenterCircumcenter

Problem Statement

Let

ABC

be an acute triangle with orthocenter

H

. Let

A_1

,

B_1

and

C_1

be the feet of the altitudes of triangle

ABC

opposite to vertices

A

,

B

, and

C

respectively. Let

B_2

and

C_2

be the midpoints of

BB_1

and

CC_1

, respectively. Let

O

be the intersection of lines

BC_2

and

CB_2

. Prove that

O

is the circumcenter of triangle

ABC

if and only if

H

is the midpoint of

AA_1

.
Proposed by Dorlir Ahmeti

Back to Problems View on AoPS