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.
Solution
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 .
Solution 2
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.
1977 Canadian MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 3 |