Constructing Inconic from Pair of Isogonal Conjugates

by AlastorMoody, Jun 5, 2019, 2:52 PM

Lemma 1: Let $P$ be a point outside ellipse $\mathcal{E}$ (ellipse), then, if $PX$ & $PY$ are tangents to $\mathcal{E}$ from $P$ and $F_1$, $F_2$ are focii of $\mathcal{E}$, then, $F_1P$, $F_2P$ are isogonal WRT $\angle XPY$
Proof: Using the Optical Property of Ellipse, $P$ is $F_1-$excenter WRT $\Delta F_1XY$ $\qquad \blacksquare$
Lemma 2: The Complement of Nagel Point is the Incenter
Proof: Let $(I)$ touch $BC$ at $D$, Let $AE_A$ be isotomic of $AD$, $E_A \in BC$. Let $AE_A$ $\cap$ $(I)$ $=$ $E$ $\implies$ $D-I-E$, using USAMO 2001 P2 and the fact that $IM_A || AE_A$ $\implies$ $IM_A$ $=$ $\frac{1}{2}ANa$ $\implies$ $I-G-Na$ and $I$ is the complement of $Na$ $\qquad \blacksquare$
Problem: 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 $P,Q$
Proof: (By ElevenCubed)
ElevenCubed wrote:
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. Hence $R$ is the center of $\Omega$ and we are done.
This post has been edited 10 times. Last edited by AlastorMoody, Aug 12, 2020, 5:13 AM

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  • what a goat, u used to be friends with my brother :)

    by bookstuffthanks, Jul 31, 2024, 12:05 PM

  • hello fellow moody!!

    by crazyeyemoody907, Oct 31, 2023, 1:55 AM

  • @below I wish I started earlier / didn't have to do JEE and leave oly way before I could study conics and projective stuff which I really wanted to study :( . Huh, life really sucks when u are forced due to peer pressure to read sh_t u dont want to read

    by kamatadu, Jan 3, 2023, 1:25 PM

  • Lots of good stuffs here.

    by amar_04, Dec 30, 2022, 2:31 PM

  • But even if he went to jee he could continue with this.

    Doing JEE(and completely leaving oly) seems like a insult to the oly math he knows

    by HoRI_DA_GRe8, Feb 11, 2022, 2:11 PM

  • Ohhh did he go for JEE? Good for him, bad for us :sadge:. Hmmm so that is the reason why he is inactive
    Btw @below finally everyone falls to the monopoly of JEE :) Coz IIT's are the best in India.

    by BVKRB-, Feb 1, 2022, 12:57 PM

  • Kukuku first shout of 2022,why did this guy left this and went for trashy JEE

    by Commander_Anta78, Jan 27, 2022, 3:42 PM

  • When are you going to br alive again ,we miss you

    by HoRI_DA_GRe8, Aug 11, 2021, 5:10 PM

  • kukuku first shout o 2021

    by leafwhisker, Mar 6, 2021, 5:10 AM

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    by WolfusA, Sep 24, 2020, 7:58 PM

  • nice blog :)

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  • Hello everyone, nice blog :)

    by Functional_equation, Sep 12, 2020, 6:22 PM

  • pro blogo

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