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Line perpendicular to the Euler line(Again!) with a beautiful concurrency

Source: STEMS 2022 Math Cat A/B P5

January 24, 2022

geometry

Problem Statement

Let

\triangle ABC

be an acute angled triangle. Let

G

be the centroid and let

D

be the foot of the altitude from

A

onto

BC

. Let ray

GD

intersect

(ABC)

at

X

and let

Z

intersect nine point circle at

Y

not on

DY

. Let

Z

be the intersection of the

\text{A-tangent}

to

(ABC)

and

\text{A-midline}

. Prove that perpendicular from

Z

to the Euler line,

AX

and

DY

concur.
The line joining the midpoints of

AB

and

AC

is called the

\text{A-midline}

.

(ABC)

denotes the circumcircle of

\triangle ABC

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