MathDB

Let

ABC

and

A'B'C'

be two triangles so that the midpoints of

\overline{AA'}, \overline{BB'}, \overline{CC'}

form a triangle as well. Suppose that for any point

X

on the circumcircle of

ABC

, there exists exactly one point

X'

on the circumcircle of

A'B'C'

so that the midpoints of

\overline{AA'}, \overline{BB'}, \overline{CC'}

and

\overline{XX'}

are concyclic. Show that

ABC

is similar to

A'B'C'

.
Proposed by usjl

1