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Figure of midpoints and feet of altitude triangle

Source: VMO 2019

April 15, 2019

geometryconcurrencyConcyclic

Problem Statement

Let

ABC

be an acute, nonisosceles triangle with inscribe in a circle

(O)

and has orthocenter

H

. Denote

M,N,P

as the midpoints of sides

BC,CA,AB

and

D,E,F

as the feet of the altitudes from vertices

A,B,C

of triangle

ABC

. Let

K

as the reflection of

H

through

BC

. Two lines

(O)

meet at

X

; two lines

DF,MN

meet at

Y

. a) The line

XY

cut the minor arc

BC

of

(O)

at

Z

. Prove that

K,Z,E,F

are concyclic. b) Two lines

KE,KF

cuts

(O)

second time at

S,T

. Prove that

BS,CT,XY

are concurrent.

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