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What if radii met heights?

Source: 2024 IRN-SGP-TWN Friendly Math Competition P5

August 2, 2024

geometry

Problem Statement

Let

ABC

be a triangle and

H, O

be its orthocenter and circumcenter, respectively. Construct a triangle by points

D_1, E_1, F_1,

where

D_1

lies on lines

BO

and

AH

,

E_1

lies on lines

CO

and

BH

, and

F_1

lies on lines

AO

and

CH

. On the other hand, construct the other triangle

D_2E_2F_2

that

D_2

lies on

CO

and

AH

,

E_2

lies on

AO

and

BH

, and

F_2

lies on lines

BO

and

CH

. Prove that triangles

D_1E_1F_1

and

D_2E_2F_2

are similar.
Proposed by Saintan Wu

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