Spring 2010 Problem 5: Circles and lines
In the image below, lines of equal length have been drawn from the point C to points X and Y on the circle. The rays CX and CY are tangent to the circle, and the line segments CX and CY both have a length of 10. A point P is randomly chosen on the circle between X and Y, and the line tangent to the circle at P is drawn. Where that line intersects the rays, the points A and B are determined. What is the perimeter of triangle ABC?
Solution
Charles Madden submitted the only answer (and the only correct answer) of 20, earning two points for the answer and one point for the first correct answer. Imagine the center of the circle is marked as O. Note that OX is perpendicular to CX (and ditto for OY and CY and for AB and OP). So if you look at the quadrilateral OXAP, it's a kite (that is, it's formed by two pairs of non-overlapping congruent sides). So the lengths of AP and AX are equal. Thus the sum of the lengths of CA and AP equals the length of CY (which is given as 10). Following the same reasoning for the other side of the triangle, you get a total perimeter of 20.