Miquel's Triangle Theorem states that if points a, b, and c are chosen on the sides of triangle ABC, the circles defined by bAc, cBa, and aCb will intersect at a common point. If a, b, and c are collinear, this point of intersection will lie on the circle defined by triangle ABC. The point of intersection is known as the 'Miquel Point'.
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1. 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 picture shows Miquel’s Theorem in action. Changing the size or shape of triangle ABC, or moving any of the side points a, b or
c, will move but not destroy the point of intersection of the three red circles. The magenta triangle abc reduces to a line when its vertices are
collinear, as on the right, and at this point we find the red circles intersect in a point 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: http://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 Amer-
ica, 1996.
Theorem of the Day is maintained by Robin Whitty at www.theoremoftheday.org