Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.9 AIME, harder Prove it

9) (14 points) $P_{0}$ is any point outside the line $l$ , for any three points $M_{1} 、 M_{2} 、 M_{3}$ on $l$ , if the circumcenters of $\triangle P_{0} M_{2} M_{3} 、 \triangle P_{0} M_{1} M_{3} 、 \triangle P_{0} M_{1} M_{2}$ are $P_{1} 、 P_{2} 、 P_{3}$ respectively. Prove: $P_{0}$ 、 $P_{1} 、 P_{2} 、 P_{3}$ are concyclic.

Solution

(9) As shown in the figure, $P_{2}$ and $P_{3}$ lie on the perpendicular bisector of $P_{0} M_{1}$ , and $P_{1}$ and $P_{2}$ lie on the perpendicular bisector of $P_{0} M_{3}$ . Let $E$ and $F$ be the feet of the perpendiculars. Note that $\angle P_{0} P_{3} E = \angle P_{0} M_{2} M_{1}$ , and $\angle P_{0} P_{1} F$ is supplementary to $\angle P_{0} M_{2} M_{3}$ . Therefore, $\angle P_{0} P_{3} E = \angle P_{0} P_{1} F$ , so $P_{0}$ , $P_{1}$ , $P_{2}$ , and $P_{3}$ are concyclic.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.