Miquel's Triangle Theorem

THEOREM OF THE DAY

Miquel's Triangle Theorem Let A, B and C be the vertices of a triangle and a, b and c be points chosen on sides CB, AC and AB, respectively. Then the circles defined by bAc, cBa and aCb have a common point of intersection. Moreover, if a, b and c are chosen to be collinear then this point lies on the circle defined by A, B and C.

The above applet, created using David Joyce's Geometry Applet software, shows Miquel's Theorem in action. Click and drag any of the points (A, B, C, a, b or c ) to move the point of intersection of the three red circles. The magenta triangle abc reduces to a line when its vertices are collinear and at this point you should find the red circles meet in a point lying on the green circle on ABC.

Auguste Miquel was a French mathematician active in the mid-nineteenth century. The point of intersection of the circles in this theorem is known as the 'Miquel Point'.

Web link: www-math.mit.edu/~kedlaya/geometryunbound/geom-080399.pdf (360K, see section 1.2)

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

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