# Kimberling’s point X(22)

## Exeter point X(22)

Exeter point is the perspector of the circummedial triangle and the tangential triangle By another words, let be the reference triangle (other than a right triangle). Let the medians through the vertices meet the circumcircle of triangle at and respectively. Let be the triangle formed by the tangents at and to (Let be the vertex opposite to the side formed by the tangent at the vertex A). Prove that the lines through and are concurrent, the point of concurrence lies on Euler line of triangle the point of concurrence lies on Euler line of triangle where - circumcenter, - orthocenter, - circumradius.

**Proof**

At first we prove that lines and are concurrent. This follows from the fact that lines and are concurrent at point and * Mapping theorem*.

Let and be the midpoints of and respectively. The points and are collinear. Similarly the points and are collinear.

Denote the inversion with respect It is evident that

Denote

The power of point with respect is

Similarly the power of point with respect is

lies on radical axis of and

Therefore second crosspoint of and point lies on line which is the Euler line of as desired.

Last we will find the length of as desired.

**Mapping theorem**

Let triangle and incircle be given. Let be the point in the plane Let lines and crossing second time at points and respectively.

Prove that lines and are concurrent.

**Proof**

We use Claim and get: Similarly,

We use the trigonometric form of Ceva's Theorem for point and triangle and get We use the trigonometric form of Ceva's Theorem for triangle and finish proof that lines and are concurrent.

**Claim (Point on incircle)**

Let triangle and incircle be given. Prove that

**Proof**

Similarly

We multiply and divide these equations and get:

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