2007 Indonesia MO Problems/Problem 7
Points are on circle , such that is the diameter of , but is not the diameter. Given also that and are on different sides of . The tangents of at and intersect at . Points and are the intersections of line with line and line with line , respectively.
(a) Prove that , , and are collinear.
(b) Prove that is perpendicular to line .
Let be the center of the circle. Let and . To prove that is on line , we can show that .
Because is a diameter, we know that and . By the Vertical Angle Theorem, , so by AA Similarity, . Thus, . Because , by SAS Similarity, , so . Since and are both inscribed angles with the same arc of a given circle, , so .
Because and are both tangents to the circle, we must have . Therefore, and . Additionally, from a property of inscribed angles, we must have and . Thus, since the sum of the angles in a triangle is , .
Additionally, since is a right triangle, we must have . Because and , we know that are in a circle with center , so . Since is a line, , so by the Base Angle Theorem, . Thus, from the Angle Addition Postulate, .
Thus, we proved that is on line . Additionally, by letting be the intersection of and , we must have and , so . By definition, , and since is on line , .
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