2013 USAJMO Problems/Problem 5
Quadrilateral is inscribed in the semicircle with diameter . Segments and meet at . Point is the foot of the perpendicular from to line . Point lies on such that line is perpendicular to line . Let be the intersection of segments and . Prove that
Let us use coordinates. Let O, the center of the circle, be (0,0). WLOG the radius of the circle is 1, so set Y (1,0) and X (-1,0). Also, for arbitrary constants and set A and B . Now, let's use our coordinate tools. It is easily derived that the equation of is and the equation of is , where and are defined appropriately. Thus, by equating the y's in the equation we find the intersection of these lines, , is . Also, . It shall be left to the reader to find the slope of , the coordinates of Q and C, and use the distance formula to verify that .
First of all , since the quadrilateral is cyclic, and triangle is rectangle, and is orthogonal to . Now because is cyclic and we have proved that , so is parallel to , and , . Now by Ptolomey's theorem on , we have , we see that triangles and are similar since and , already proven, so , substituting we get , dividing by , we get . Now triangles , and are similar so , but also triangles and are similar and we get , comparing we have, substituting, . Dividing the new relation by and multiplying by we get , but , since triangles and are similar, because and since . Substituting again we get . Now since triangles and are similar we have and by the similarity of and , we get so substituting, and separating terms we get , but in the beginning we prove that and so , and we are done.