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Question Number 87222 by TawaTawa1 last updated on 03/Apr/20

Commented by TawaTawa1 last updated on 03/Apr/20

$$\mathrm{Circles}{\omega }_1\mathrm{and}{\omega }_2\mathrm{intersect}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{points}\mathrm{A}\mathrm{and}\mathrm{B}.$$$$\mathrm{Point}\mathrm{C}\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{tangent}\mathrm{line}\mathrm{from}\mathrm{A}\mathrm{to}{\omega }_1\mathrm{such}\mathrm{that}$$$$\mathrm{\angle }\mathrm{ABC}=90°.\mathrm{Arbitrary}\mathrm{line}\mathrm{L}\mathrm{passes}\mathrm{through}\mathrm{C}\mathrm{and}$$$$\mathrm{cuts}{\omega }_2\mathrm{at}\mathrm{points}\mathrm{P}\mathrm{and}\mathrm{Q}.\mathrm{Lines}\mathrm{AP}\mathrm{and}\mathrm{AQ}\mathrm{cut}{\omega }_1$$$$\mathrm{for}\mathrm{the}\mathrm{second}\mathrm{time}\mathrm{at}\mathrm{points}\mathrm{X}\mathrm{and}\mathrm{Z}\mathrm{respectively}.\mathrm{Let}$$$$\mathrm{Y}\mathrm{be}\mathrm{the}\mathrm{foot}\mathrm{of}\mathrm{altitude}\mathrm{from}\mathrm{A}\mathrm{to}\mathrm{L}.$$$$\mathrm{Prove}\mathrm{that}\mathrm{points}\mathrm{X},\mathrm{Y}\mathrm{and}\mathrm{Z}\mathrm{are}\mathrm{collinear}.$$

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