1977 Canadian MO Problems/Problem 2
Let be the center of a circle and be a fixed interior point of the circle different from Determine all points on the circumference of the circle such that the angle is a maximum.
If is the chord perpendicular to through point , then extend to meet the circle at point . It is now evident that is the midpoint of , is the midpoint of , and hence .
Similarly, let be a point on arc . Extend to meet the circle at point . Extend to meet the circle a second time at .
We now plot on such that . Then, . Since , . Hence, , and therefore, .
Ergo, the points such that is maximized are none other than points and .
Let be the chord perpendicular to through . We claim that the choices of that maximize are and .
Let be the circumcircle of . Then clearly is maximized if the radius of is minimized, which occurs if is internally tangent to circle . Hence is a diameter of (whose center is collinear with), and so . It follows that or , as desired.
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