Some Properties of the Orthic Triangle - Conics

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Some Properties of the Orthic Triangle - Conics

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#1 h Mar 28, 2014, 4:49 PM • 2 Y

Y by Adventure10, Mango247

Some conics associated to the orthic triangle.

http://geometriadominicana.blogspot.com/2014/03/some-properties-of-orthic-triangle.html

Emmanuel.

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#2 h Mar 29, 2014, 1:53 PM • 2 Y

Y by Adventure10, Mango247

Here is a proof, Ill also post it as a comment

Nice site BTW!!!

Note the focii of the parabolas are $B', A', C'$ , since they are miquel points of $BAHA'C'C$ etc... (well known that focus will be in circle $AC'H$ etc...) Also, since $B'HC'A$ are concyclic, we have $C'(C, B', A, A') \cap \odot B'HC'A \implies B'HA$ are harmonic with $A'C' \cap \odot B'HC'A$ , which easily means it is the reflection of $B$ on $AA'$ , so the simson line of $B'$ wrt $\odot AA'B$ is $A'C'$ , thus it is the directrix of the parabola tangent to $AH, AB_1, CH, CA_2$ , name it $\Gamma$ . $AH, CH \cap \Gamma = X, Y$ . Note by newtons theorem, or degenerate brianchon, $XA_2 \cap YB_2 \cap AC \cap BH = B'$ , and the polars of $XA_2, YB_2$ are $A', C' \implies$ the polar of $B'$ wrt $\Gamma$ is $A'C'$ . Thus, $XB_2 \cap YA_2 = HB' \cap A'C' = B_3$ , and $XY \cap A_2B_2 = A'C' \cap AC = B_4$ . Now, the perpendicular from $B_a$ to $A'C'$ is concyclic with $B'B_aB_3$ and has equal length with $B_aB'$ . Thus, $B_3B_a$ bisects $\angle B_4B_3B'$ and similarly $BB_c$ externally bisects this angle. Thus, $(B_4, B', B_a, B_c) = -1$ , so the midpoint of $B_aB_c$ coincides with the midpoint $M_B$ of $AC$ . Hence, the result will follow from this easy lemma:

For a triangle $ABC$ , let $A'B'C'$ be point on the sides of it, and let the isotomic points of them be $A''B''C''$ respectively. Then, $A'A''B'B''C'C''$ lie on a conic.

I am yet to find a synthetic proof, the result however is trivial by Carnot's theorem on conics.

Nice result!

This post has been edited 1 time. Last edited by IDMasterz, Mar 31, 2014, 6:14 AM

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#3 h Mar 29, 2014, 2:06 PM • 2 Y

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Thank you, IDMasterz.

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