MathDB

Let

ABC

be an acutangle scalene triangle with

\angle BAC = 60^{\circ}

and orthocenter

H

. Let

\omega_b

be the circumference passing through

H

and tangent to

AB

B

, and

\omega_c

the circumference passing through

H

and tangent to

AC

C

.
[*] Prove that

\omega_b

and

\omega_c

only have

H

as common point. [*] Prove that the line passing through

H

and the circumcenter

O

of triangle

ABC

is a common tangent to

\omega_b

and

\omega_c

.
Note: The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

Problem Statement