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Let points

A, B, C, A', B', C'

be chosen in the plane such that no three of them are collinear, and let lines

AA'

BB'

and

CC'

be tangent to a given equilateral hyperbola at points

A

B

and

C

, respectively. Assume that the circumcircle of

A'B'C'

is the same as the nine-point circle of triangle

ABC

. Let

s(A')

be the Simson line of point

A'

with respect to the orthic triangle of

ABC

. Let

A^*

be the intersection of line

B'C'

and the perpendicular on

s(A')

from the point

A

. Points

B^*

and

C^*

are defined in a similar manner. Prove that points

A^*

B^*

and

C^*

are collinear.
Submitted by Áron Bán-Szabó, Budapest

Problem Statement