1990 USAMO Problems/Problem 5
An acute-angled triangle is given in the plane. The circle with diameter intersects altitude and its extension at points and , and the circle with diameter intersects altitude and its extensions at and . Prove that the points lie on a common circle.
Let be the intersection of the two circles (other than ). is perpendicular to both , implying , , are collinear. Since is the foot of the altitude from : , , are concurrent, where is the orthocentre.
Now, is also the intersection of , which means that , , are concurrent. Since , , , and , , , are cyclic, , , , are cyclic by the radical axis theorem.
Define as the foot of the altitude from to . Then, is the orthocenter. We will denote this point as . Since and are both , lies on the circles with diameters and .
Now we use the Power of a Point theorem with respect to point . From the circle with diameter we get . From the circle with diameter we get . Thus, we conclude that , which implies that , , , and all lie on a circle.
Solution 3 (Radical Lemma)
Let be the circumcircle with diameter and be the circumcircle with diameter . We claim that the second intersection of and other than is , where is the feet of the perpendicular from to segment . Note that so lies on Similarly, lies on . Hence, is the radical axis of and . By the Radical Lemma, it suffices to prove that the intersection of lines and lie on . But, is the same line as and is the same line as . Since , and intersect at the orthocenter , lies on the radical axis and we are done.
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