Tessellate S.T.E.M.S. (2019) - Mathematics - Category B - Set 2 - Subjective Problem 2

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ intersect at points $A$ and $B$ . A line through $A$ intersects the circles $\omega_1$ and $\omega_2$ at $Y$ and $Z$ respectively. Let the tangents at $Y$ and $Z$ intersect at $X$ and lines $YO_1$ and $O_2$ intersect at $P$: Let the circumcircle of $\Delta O_1O_2B$ have it's center at $O$ and intersect line $XB$ at $B$ and $Q$. Prove that $PQ$ is a diameter of the circumcircle of $\Delta O_1O_2B$.

This problem is a part of Tessellate S.T.E.M.S. (2019)

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$