Eight Point Circle: What is it?
A Mathematical Droodle
Explanation
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Copyright © 1996-2012 Alexander Bogomolny
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Assume the diagonals AC and BD of a quadrilateral ABCD are orthogonal. Then the midpoints P, Q, R, S and the feet E, F, G, H of the perpendiculars from the midpoints to the opposite sides all lie on a circle centered at the gravity center K of ABCD.
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K is the center of the Varignon parallelogram, which, since the diagonals of ABCD are orthogonal, is a rectangle. It follows that segments PR and QS serve as diameters of a circle with center at K. But then right angles subtended by either PR or QS are inscribed into that circle. This, in particular, includes points E, F, G and H.
The circle, now known as the 8-point circle, was identified in as late as 1944 by Louis Brand of Cincinnati.
Simple as it is, in a triangle, existence of the 8-eight point circle leads to a by far more famous 9 point circle.
References
- R. Honsberger, Mathematical Gems, II, MAA, 1976
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Copyright © 1996-2012 Alexander Bogomolny
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