# Simson line

In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear.

## Simson line (main)

Let a triangle and a point be given.

Let and be the foots of the perpendiculars dropped from P to lines AB, AC, and BC, respectively.

Then points and are collinear iff the point lies on circumcircle of

**Proof**

Let the point be on the circumcircle of

is cyclic

is cyclic

is cyclic

and are collinear as desired.

**Proof**

Let the points and be collinear.

is cyclic

is cyclic

is cyclis as desired.

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

## Simson line of a complete quadrilateral

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

Let be the Miquel point of a complete quadrilateral.

Let and be the foots of the perpendiculars dropped from to lines and respectively.

Prove that points and are collinear.

**Proof**

Let be the circumcircle of be the circumcircle of Then

Points and are collinear as Simson line of

Points and are collinear as Simson line of

Therefore points and are collinear, as desired.

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

## Problem

Let the points and be collinear and the point

Let and be the circumcenters of triangles and

Prove that lies on circumcircle of

**Proof**

Let and be the midpoints of segments and respectively.

Then points and are collinear

is Simson line of lies on circumcircle of as desired.

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