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H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E+5

May 25, 2024

geometry

Problem Statement

Consider an acute triangle

ABC

. Let

D

,

E

and

F

be the feet of the altitudes through vertices

A

,

B

and

C

. Denote by

A'

,

B'

,

C'

the projections of

A

,

B

,

C

onto lines

EF

,

FD

,

DE

, respectively. Further, let

H_D

,

H_E

,

H_F

be the orthocenters of triangles

DB'C'

,

EC'A'

,

FA'B'

. Show that

H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.

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