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## Re: [EMHL] Three Circles Around Gergonne Point

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• Dear Quang! ... circles taken GeA , GeB , GeC as diameters cut incircle at A B C respectively (other than A B C ). Three lines AA , BB , CC cut incircle
Message 1 of 1 , Sep 29, 2006
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Dear Quang!

> Given triangle ABC, Gergonne point Ge, intouch triangle A'B'C'. Three
circles taken GeA', GeB', GeC' as diameters cut incircle at A''B''C''
respectively (other than A'B'C'). Three lines AA', BB', CC' cut incircle at
A*, B*, C* respectively (other than A'B'C').
> Result:
> Two triangles A''B''C'', A*B*C* are perspective with perspector P not in
current ETC.
> Barycentrics of P:
> a + 2*b + 2*c - 8*b*c / (b + c - a) : :
> Search value: + 0.340522621678326
>
This is a partial case of next fact.
Let the circle with center O be given. P is fixed point inside this circle.
For any point X on the circle let X_1 be the second common point of the
circle and the line PX; X_2 be the second common point of given circle and
the circle with diameter PX. Then all lines X_1X_2 pass through the fixed
point P'. In Klein model of Lobachevsky geometry P is the midpoint of OP'.
This fact can be proved using the inversion with center P.

Sincerely
Alexey

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