Question 90
Let $\triangle ABC$ be a triangle with $|AB| \lt |AC| \lt |BC|$ and circumcircle $\Gamma$ with centre $O$. Let $\omega_1$ be the circle with centre $B$ and radius $|AC|$ and let $\omega_2$ be the circle with centre $C$ and radius $|AB|$. The circles $\omega_1$ and $\omega_2$ intersect at a point $E$ such that $A$ and $E$ lie on different sides of the line $BC$. The circles $\Gamma$ and $\omega_1$ intersect at a point $F$ and the circles $\Gamma$ and $\omega_2$ intersect at a point $G$, such that $F$ and $G$ lie on the same side of the line $BC$ as $E$.
Prove that the antipode $K$ of $A$ with respect to $\Gamma$ is the centre of the circumcircle of $\triangle EFG$.
The antipode $K$ of a point $A$ is the unique point on the circle such that $AK$ is a diameter.
Prove that the antipode $K$ of $A$ with respect to $\Gamma$ is the centre of the circumcircle of $\triangle EFG$.
The antipode $K$ of a point $A$ is the unique point on the circle such that $AK$ is a diameter.