Kimberling’s point X(24)
Kimberling's point X(24)
Kimberling defined point X(24) as perspector of and Orthic Triangle of the Orthic Triangle of .
Theorem 1
Denote obtuse or acute Let be the base triangle, be Orthic triangle of be 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
Well known that is the polar of point so
Let be inversion with respect
Denote
Theorem 2
Let be the base triangle, be orthic triangle of be Kosnita triangle of 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
Proof
Let be orthocenter of be the center of Nine-point circle of is the Euler line of Well known that is antiparallel with respect
is the bisector of therefore is antiparallel with respect Similarly, and are homothetic.
Let be the center of homothety.
is -excenter of is -excenter of
Denote circumradius is the point
vladimir.shelomovskii@gmail.com, vvsss
Theorem 3
Let be the reference triangle (other than a right triangle). Let the altitudes 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 one can prove that lines and are concurrent. This follows from the fact that lines and are concurrent at point and Mapping theorem (see Exeter point
Let and be the midpoints of and respectively.
Let and be the midpoints of and respectively.
Let and be the foots of altitudes from and respectively.
The points and are collinear. Similarly the points and are collinear.
Denote the inversion with respect It is evident that
Denote
It is known that
Similarly,
We use Claim and get that the power of point with respect each circle is
lies on common radical axis of and
Therefore second crosspoint of these circles point lies on line which is the Euler line of lies on the same Euler line as desired.
Last we will find the length of as desired.
vladimir.shelomovskii@gmail.com, vvsss