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Oops, I lost my compass...

Source: USEMO 2019 Problem 6, by Ankan Bhattacharya

May 24, 2020

geometryUSEMOHi

Problem Statement

Let

X

be an acute scalene triangle with circumcenter

O

and altitudes

\overline{AD}

,

\overline{BE}

,

\overline{CF}

. Let

X

,

Y

,

Z

be the midpoints of

\overline{AD}

,

\overline{BE}

,

\overline{CF}

. Lines

AD

and

YZ

intersect at

P

, lines

BE

and

ZX

intersect at

Q

, and lines

CF

and

XY

intersect at

R

.
Suppose that lines

YZ

and

BC

intersect at

A'

, and lines

QR

and

EF

intersect at

D'

. Prove that the perpendiculars from

A

,

B

,

C

,

O

, to the lines

QR

,

RP

,

PQ

,

A'D'

, respectively, are concurrent.
Ankan Bhattacharya

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