Mikel's theorem is a theorem about intersecting circles in geometry. In 1838, August Mick described and proved several related theorems. Many useful theorems can be deduced from it.

Theorem statement

Three-circle theorem: Let three circles of C 1, C2, C3 and C3 intersect at one point, o, m, n, P N and p are other intersections of C 1 and C2, C2 and C3, C3 and C 1 respectively. Let A be the point of C 1, the straight line MA passes through C2 in B, the straight line PA passes through C3 in C ... and then the three lines of B, N and c * * *. Inverse theorem: If △ABC is a triangle, and M, N and P are on the sides of AB, BC and CA respectively, then the circumscribed circles of △ amp, △ BMN and △ CNP intersect at a point O.

Four-circle theorem: let C 1, C2, C3 and C4 be four circles, A 1 and B 1 are the intersections of C 1 and C2, A2 and B2 are the intersections of C2 and C3, A3 and B3 are the intersections of C3 and C4, and A4 and B4 are C/kloc. Then A 1, A2, A3 and A4 are * * * cycles if and only if B 1, B2, B3 and B4 are * * * cycles. Five-circle theorem: Let ABCDE be an arbitrary pentagon, and five points F, G, H, I and J are the intersections of EA and BC, AB and CD, BC and DE, CD and EA, DE and AB, then the five intersections of the circumscribed circle of the triangle △ABF, △BCG, △CDH, △DEI, △EAJ are not on the pentagon * *.

Inverse Theorem: Let the centers of five circles C 1, C2, C3, C4 and C5 all lie on the circle C, and the adjacent circles intersect on the circle C, then connect their intersections not on the circle C with their adjacent points, and five straight lines intersect on the five circles.

1838, August Meeker published a part of this theorem in joseph liouville's Journal de Mathé matiques Pures et Appliqué es (Journal of Pure and Applied Mathematics).

Mick's first theorem is a famous classic result that existed a long time ago, and it is proved by the theorem of circle angle.

The intersection of four circles on a perfect quadratic curve is now called the Mikel point, but this property was known by Jacob Steiner in 1828, and probably also by william wallace.

The five-circle theorem is a special case of a more general theorem. This definition was put forward and proved by william king Deng Clifford.