MathDB

Two circles

\Gamma_1

and

\Gamma_2

intersect at points

A

and

Z

(with

A\neq Z

). Let

B

be the centre of

\Gamma_1

and let

C

be the centre of

\Gamma_2

. The exterior angle bisector of

\angle{BAC}

intersects

\Gamma_1

again at

X

and

\Gamma_2

again at

Y

. Prove that the interior angle bisector of

\angle{BZC}

passes through the circumcenter of

\triangle{XYZ}

.
For points

P,Q,R

that lie on a line

\ell

in that order, and a point

S

not on

\ell

, the interior angle bisector of

\angle{PQS}

is the line that divides

\angle{PQS}

into two equal angles, while the exterior angle bisector of

\angle{PQS}

is the line that divides

\angle{RQS}

into two equal angles.

Problem Statement