MathDB

Let

D

be the foot of perpendicular from

A

to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle

ABC

. A circle

\omega

with centre

S

passes through

A

and

D

, and it intersects sides

AB

and

AC

X

and

Y

respectively. Let

P

be the foot of altitude from

A

BC

, and let

M

be the midpoint of

BC

. Prove that the circumcentre of triangle

XSY

is equidistant from

P

and

M

Problem Statement