Problem:
In the adjoining figure, two circles of radii and are drawn with their centers units apart. At , one of the points of intersection, a line is drawn in such a way that the chords and have equal length. Find the square of the length of .
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Solution:
We complete the figure as shown. Applying the Pythagorean Theorem to the shaded trΔ±angle gives
But and . Thus
Simplifying gives . Squaring and simplifying gives . Thus .
The problems on this page are the property of the MAA's American Mathematics Competitions