Relative Positions Of Two Circles

page revision: 0, last edited: 15 Sep 2008 09:00

Relative Positions Of Two Circles

page revision: 0, last edited: 15 Sep 2008 09:00

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As it was pointed to me by Cathy Gray and independently by Thomas Foregger, there is a serious issue with Problem 254.

It was overlooked by all editors of the book starting from its first Russian edition of 1892.

Apparently, a different problem was meant to be here:

Prove that the greatest among segments connecting a point of one circle with a point of another lies on the line of centers.The formulation given in the book is

incorrect.It can be corrected this way:Through a point of intersection of two circles, secants are drawn which meet the circles at the points A and B. Prove that the segment AB is the greatest when the secant is parallel to the line of centers.When the point C of intersection lies between A and B, this segment AB coincides with the whole secant described in the book. However when A and B lie on one side of C, the segment AB is only a part of the whole secant. Depending on the mutual position of the circles, such a whole secant may indeed become greater than the one parallel to the line of centers, thus making the statement given in the book false.

In the corrected formulation, the proof known to me is based on similarity. The corrected problem should therefore be placed into the section "Similarity of Triangles." Even there, the problem still merits the asterisk indicating a higher difficulty level.

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