1975 USAMO Problems/Problem 4
Two given circles intersect in two points and . Show how to construct a segment passing through and terminating on the two circles such that is a maximum.
A maximum cannot be attained if intersects segment because a larger value can be attained by making one of or diametrically opposite , which (as is easily checked) increases the value of both and . Thus, assume does not intersect .
Let and be the centers of the small and big circles, respectively, and and be their respective radii.
Let and be the feet of and to , and and
is maximum when the product is a maximum.
But and is fixed, so is .
So its maximum depends on which occurs when . To draw the line :
Draw a circle with center and radius to cut the radius at . Draw the line parallel to passing through . This line meets the small and big circles at and , respectively.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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