Kimberling’s point X(24)
Kimberling point X(24)
Perspector of Triangle and Orthic Triangle of the Orthic Triangle. Theorem 1 Denote obtuse or acute Let be the base triangle, be Orthic triangle of be Orthic Triangle of the Orthic Triangle of . Let and be the circumcenter and orthocenter of
Then and are homothetic, the point center of this homothety lies on Euler line of
The ratio of the homothety is
Proof
WLOG, we use case Let be reflection in
In accordance with Claim, and are collinear.
Similarly, and were is reflection in are collinear.
Denote
and are concurrent at point
In accordance with Claim, points and are isogonal conjugate with respect
Claim
Let be an acute triangle, and let and denote its altitudes. Lines and meet at Prove that
Proof
Let be the circle centered at is midpoint
Let meet at Let be the circle centered at with radius
Let be the circle with diameter
We know that
Let be inversion with respect
Denote
Theorem 2
Let be the base triangle, be orthic triangle of be Kosnita triangle. Then and are homothetic, the point center of this homothety lies on Euler line of the ratio of the homothety is We recall that vertex of Kosnita triangle are: is the circumcenter of is the circumcenter of is the circumcenter of where is circumcenter of
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