Perspector of Inconic from Isogonal Conjugates

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Perspector of Inconic from Isogonal Conjugates

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#1 h Jun 2, 2019, 2:20 PM • 3 Y

Y by ElevenCubed, Adventure10, Mango247

Let $(P,Q)$ be a pair of Isogonal Conjugates WRT $\Delta ABC$ . Let $R$ be the midpoint of $PQ$ . Let $M$ be the isotomic conjugate of the anticomplement of $R$ . If $\Delta DEF$ is the cevian triangle of $M$ WRT $\Delta ABC$ , then Prove, that $D,E,F$ are the tangency points of the inconic with focii $P,Q$

This post has been edited 1 time. Last edited by AlastorMoody, Jun 2, 2019, 2:21 PM

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#2 h Jun 5, 2019, 2:31 PM • 4 Y

Y by AlastorMoody, Pluto1708, Adventure10, Mango247

Let $\Omega$ be the inconic tangent at $D, E$ and $F$ . It enough to prove, that $R$ is the center of this conic, as it is well known that the focii will be isogonal conjugates, and there is a unique conic tangent to the sides of the triangle, given it's center.

We can now make $\Omega$ into a circle, with an affine transformation. $M$ becomes the Gergonne point, so $R$ becomes the Nagel point as it is still the isotomic conjugate of $M$ . The anticomplement $R$ of the Nagel point is the incenter (is this well known?). Hence $R$ is the center of $\Omega$ and we are done.

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#3 h Jun 5, 2019, 2:48 PM • 1 Y

Y by Adventure10

@above thanks! and yes, anticomplement of Nagel Point is Incenter is really well-known

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#4 h Jun 5, 2019, 2:56 PM • 2 Y

Y by AlastorMoody, Adventure10

Aha, thanks

Clarification:
Also, with complex affine transformations, we can make any conic into a circle, so it also holds if $R$ is outside the triangle.

This post has been edited 2 times. Last edited by ElevenCubed, Jun 5, 2019, 3:08 PM
Reason: to fit context

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