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
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.
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:
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