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Orthic triangle DEF satisfying 3EF=FD+DE

Source: Centroamerican 2011, Problem 6

June 22, 2011

trigonometrygeometryperpendicular bisectorgeometry proposed

Problem Statement

Let

ABC

be an acute triangle and

D

,

E

,

F

be the feet of the altitudes through

A

,

B

,

C

respectively. Call

Y

and

Z

the feet of the perpendicular lines from

B

and

C

to

FD

and

DE

, respectively. Let

F_1

be the symmetric of

F

with respect to

E

and

E_1

be the symmetric of

E

with respect to

F

. If

3EF=FD+DE

, prove that

\angle BZF_1=\angle CYE_1

.

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