viernes, 26 de octubre de 2012

A conic centered at the Euler line

Theorem. Given a triangle $ABC$, call $\Gamma$ the locus of points $P$ such that polar of $P$ with respect to the circumcircle is tangent to the nine point circle. Then we have:
1) $\Gamma$ is a conic whose center is $X_{26}$, the circumcenter of the tangential triangle.
2) $\Gamma$ is an ellipse, parabola o hyperbola if and only if the triangle is acute, rectangle or obtuse.
3) The diameter on the transverse axis of the conic is also a diameter of the circumcircle of tangential triangle. Therefore, the transverse axis of the conic is the Euler line of the triangle.
4) If  $\alpha$, $\beta$  are the lengths of the transverse and conjugate axes of the conic respectively, we have the relation

                                $\left(\frac{\alpha}{\beta}\right)^2=1-\frac{OH^2}{R^2}$

5) The foci of the conic are $O$ and $O'$, where $O'$ is the reflection of $O$ on the center $X_{26}$.

In fact, this is a particular case of some projective theorem: "the locus of poles with respect a conic of the tangents to another conic is also a conic".

Here is the version for two circles:

$(A)$ and $(B)$ are circles
The line $AB$ intersect $(B)$ at $M$ and $N$
$M'$ and $N'$ are the inverses of $M$ and $N$ with respect to $(A)$
$J$ is the inverse of $A$ with respect to $(B)$
$O$ is the inverse of $J$ with respect to $(A)$
$A'$ is the reflection of $A$ on $O$
The locus points $P$ such that the polar of $P$ with respect to $(A)$ is tangent to $(B)$ is a conic with foci $A$ and $A'$ and diameter $M'N'$.


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