The special case of Miquel's theorem when points are collinear | plane geometry | intermediate level

Episode 121.

The special case of Miquel's theorem when points are collinear | plane geometry | intermediate level.
The special case of Miquel's theorem when the points are collinear | plane geometry | intermediate level.

Branch of mathematics: plane geometry.
Difficulty level: intermediate.

Miquel's theorem. Let $ABC$ be a triangle. Let $D$, $E$, $F$ be arbitrary points on the lines containing the sides $BC$, $CA$, $AB$ of the triangle $ABC$. Then the circumcircles of the triangles $AEF$, $BFD$, $CDE$ intersect at a single point $M$, which is called the Miquel's point.

Special case. The Miquel's point $M$ lies on the circumcircle of the triangle $ABC$ if and only if the points $D$, $E$, $F$ are collinear.

Mathematics. Geometry. Plane geometry.
#Mathematics #Geometry #PlaneGeometry

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