The Five Circles Theorem

THEOREM OF THE DAY

The Five Circle Theorem Let the five sides of a pentagon ABCDE be extended until they intersect in five points P, Q, R, S and T, say. Then the five circumcircles of triangles BQA, APE, ETD, DSC and CRB intersect with each other in five distinct points, not lying on the pentagon and lying on a common circle.

Test the theorem with this applet, created using David Joyce's Geometry Applet software. Move the circle on points X, Y and Z until they lie exactly on any three distinct points of intersection of the circles, other than the points A, B, C, D and E of the pentagon. You will find that two other circle intersections are automatically included! Move the points P, Q, R, S and T to set yourself a new challenge!

Auguste Miquel taught mathematics in Nantua in the French Alps, and in Castres, where Fermat died nearly two hundred years earlier. He published this, and a number of other theorems relating to the geometry of circles, between 1838 and 1846.

Web link: www.mathsyear2000.co.uk/explorer/circles/miquels-circle-theorems/

Further reading: Episodes in Nineteenth and Twentieth Century Euclidean Geometry, by Ross Honsberger, The Mathematical Association of America, 1996.

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