A Random Isogonal Conjugate Thing

by djmathman, May 25, 2020, 12:47 AM

because I spent all day trying to write an olympiad Geo problem and this is all I got

Claim. Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be the intersection of the tangents to $\Omega$ at $B$ and $C$. Let $Q$ be the isogonal conjugate of $P$ with respect to $\triangle ABC$. Then $ABQC$ is a parallelogram.

Proof. Note that $\angle ABP = \angle ABC + \angle BAC = 180^\circ - \angle ACB$, so $\angle CBQ = \angle ACB$ and $AB\parallel CQ$. Analogously, $AC\parallel BQ$, and we may conclude. $\blacksquare$

Corollary. $AP$ is the $A$-symmedian of $\triangle ABC$.

On a related note, preparing for Geo 2 is a lot harder than I thought it would be >.<

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If you consider $P_1,P_2,P_3$ (defined similarly to $P$) and $Q_1,Q_2,Q_3$ (defined similarly to $Q$), then the tangential triangle of $P_1P_2P_3$ and the medial triangle of $Q_1,Q_2,Q_3$ are both $\triangle ABC$. This is trivial by definition, but, I guess it's somewhat cool?

Side note: It took me 5 minutes to realize why flipping a line over the interior angle bisector and the exterior angle bisector when it goes through the intersection of the two makes the same line (because this is true for any two perpendicular lines, but that wasn't obvious to me at first :D).

Also: Is there a nice formula for the distance between two isogonal conjugates? Would love one to length bash with XD.

by freeman66, May 25, 2020, 3:26 AM

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  • dj so orz :omighty:

    by Yiyj1, Mar 29, 2025, 1:42 AM

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  • orz $$\,$$

    by balllightning37, Jul 26, 2024, 1:05 AM

  • hi dj $ $ $ $

    by OronSH, Jul 23, 2024, 2:14 AM

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  • hi dj, may i have the role of contributer? :D

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  • waiting for a recap of your amc proposals for this year :D

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  • also happy late bday man! i missed it by 2 days but hope you are enjoyed it

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  • hello :)

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  • Do you have a link to your main blog that you started after graduating from high school, I couldn't find it. @dj I met you IRL at Awesome Math summer Program several years ago.

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