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