2004 Pan African MO Problems/Problem 6
Let be a cyclic quadrilateral such that is a diameter of it's circumcircle. Suppose that and intersect at , and at , and at , and let be a point on . Show that is perpendicular to if and only if is the midpoint of .
We use the notation to represent the circumcircle of for any three points , and use to represent the circumcircle of . Without loss of generality, assume that and are not parallel and that is closer to than to . In this solution, we use to represent the midpoint of and prove that .
Note that . Similarly, note that . Therefore, is a diameter of .
Define points such that . We claim that , and we prove this by showing that lies on and that it lies on .
To prove that , we angle chase: , , so that ; similarly, . Therefore, .
To prove that , we make a similar angle chase: , , so that ; similarly, we have . Therefore, .
Therefore, the angle between and is also the angle between and , which is trivially because is part of the circle with diameter , and we are done.
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