Miquel's Theorem

Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides.

Formally, let ABC be a triangle, with points A´, B´ and C´ on sides BC, AC, and AB respectively. Draw three circumcircles to triangles AB´C´, A´BC´, and A´B´C. Miquel's theorem then states that these circles intersect in a single point M, the Miquel point. In addition, the three angles MA´B, MB´C and MC´A (green in the diagram) are all equal, as are the three complementary angles MA´C, MB´A and MC´B.

The theorem (and its corollary) follow from the properties of two cyclic quadrilaterals drawn from any two of a triangle's vertices, having an edge in common as shown in the figure. Their combined angles at M (opposite A and opposite C) will be (180 - A) + (180 - C), giving an exterior angle equal to (A + C). Since (A + C) also equals (180 - B), the intersection at M, lying on the chord A´C´, must also lie on a cyclic quadrilateral passing through points B, A´, and C´. This completes the proof.

If the fractional distances of A´, B´ and C´ along sides BC (a), CA (b) and AB (c) are d_a, d_b and d_c, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by:

(with d'_a = 1 - d_a etc.)

In the case d_a = d_b = d_c = ½ the Miquel point is the circumcentre.

The theorem can be reversed to say: for three circles intersecting at M, a line can be drawn from any point A on one circle, through its intersection C´ with another to give B (at the second intersection). B is then similarly connected, via intersection at A´ of the second and third circles, giving point C. Points C, A and the remaining point of intersection, B´, will then be collinear, and triangle ABC will always pass though the circle intersections A´, B´ and C´.

This can be extended to a circle with four points. Given points, A, B, C, and D on a circle, and circles passing through each adjacent pair of points, the alternate intersections of these four circles at W, X, Y and Z then lie on a common circle. This is known as Miquel's six circles theorem.