# Miquel's point

## Contents

## Miquel and Steiner's quadrilateral theorem

Let four lines made four triangles of a complete quadrilateral. In the diagram these are

Prove that the circumcircles of all four triangles meet at a single point.

**Proof**

Let circumcircle of circle cross the circumcircle of circle at point

Let cross second time in the point

is cyclic

is cyclic

is cyclic

is cyclic and circumcircle of contain the point

Similarly circumcircle of contain the point as desired.

**vladimir.shelomovskii@gmail.com, vvsss**

## Circle of circumcenters

Let four lines made four triangles of a complete quadrilateral. In the diagram these are

Prove that the circumcenters of all four triangles and point are concyclic.

**Proof**

Let and be the circumcircles of and respectively.

In

In

is the common chord of and

Similarly, is the common chord of and

Similarly, is the common chord of and

points and are concyclic as desired.

**vladimir.shelomovskii@gmail.com, vvsss**

## Triangle of circumcenters

Let four lines made four triangles of a complete quadrilateral.

In the diagram these are

Let points and be the circumcenters of and respectively.

Prove that and perspector of these triangles point is the second (different from ) point of intersection where is circumcircle of and is circumcircle of

**Proof**

Quadrungle is cyclic

Spiral similarity sentered at point with rotation angle and the coefficient of homothety mapping to , to , to

are triangles in double perspective at point

These triangles are in triple perspective are concurrent at the point

The rotation angle to is for sides and or angle between and which is is cyclic is cyclic.

Therefore is cyclic as desired.

Similarly, one can prove that

**vladimir.shelomovskii@gmail.com, vvsss**

## Analogue of Miquel's point

Let inscribed quadrilateral and

points be given.

Prove that points and are concyclic.

**Proof**

**Corollary**

The points and are concyclic.

The points and are concyclic.

**vladimir.shelomovskii@gmail.com, vvsss**

## Six circles crossing point

Let point point be given.

Denote tangent to tangent to

Prove that the circles and have the common point.

**Proof**

Let points and are concyclic, Similarly is the Miquel point of quadrungle is tangent to Similarly, is tangent to

**vladimir.shelomovskii@gmail.com, vvsss**