Problem:
Let be an acute-angled triangle, and let and be two points on side . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Prove that points , and lie on a circle.
Solution:
Let denote the angles of . Without loss of generality, we assume that is on the segment .
We guess that is on the line through and . To confirm that our guess is correct and prove that , and lie on a circle, we start by letting be the point other than that is on the line through and , and on the circle through , and . Two applications of the Inscribed Angle Theorem yield and , from which we conclude that .
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From we have so quadrilateral is cyclic. By the Inscribed Angle Theorem, .
Finally, , from which it follows that and thus , and are concyclic.
This problem was proposed by Zuming Feng.
The problems on this page are the property of the MAA's American Mathematics Competitions