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The Longest Segment in Intersecting Circles
Given two intersecting circles. Draw a line through one of the intersection points, say, A. Measure BC, the segment of the line enclosed by the two circles. The problem is to find the direction of the line such that the segment BC is the longest. Solution
Copyright © 1996-2008 Alexander Bogomolny
Solution
The drawing is somewhat empty. Experiment with adding meaningful elements. Perhaps you will come up with the diagram on the right. Let D be the other point of intersection of the two circles. Connect D with B and C. In the triangle BCD, the angles B and C do not depend on the position of the line BC. Therefore, all triangles BCD obtained when the direction of BC changes are similar. Therefore, the longest BC corresponds to the longest DC. But DC is just a chord in a circle. And the longest chord is a diameter. Questions
The following problem is discussed in Honsberger, Mathematical Morsels, p. 126 and referred to in Nelsen, Proofs Without Words II:
SolutionIn the previous problem we used the fact that the angle P (APB) does not depend on the position of the point P. The same is of course true for the angle CPD. But the latter is defined by the difference of two arcs (on the left circle) CD and AB. Since the latter is fixed, so is the former. (Another solution comes with a dynamic illustration.)
Copyright © 1996-2008 Alexander Bogomolny |
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