Πέμπτη, 7 Φεβρουαρίου 2013

EULER LINE [Generalization]


Let ABC be a triangle, T a fixed point, AtBtCt the circumcevian triangle of T, A'B'C' the antipodal triangle of AtBtCt (ie A',B',C' are the antipodes of At, Bt,Ct).

Let P be a variable point and A"B"C" the circumcevian triangle wrt A'B'C'.

Which is the locus of P such that ABC, A"B"C" are perspective?

Antreas P. Hatzipolakis, 7 Feb. 2013

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The locus is the line OT, and the perspector Q lies on OT as well.

If OP : PT = t: 1-t, then

OQ : QT = R^2(1+t) : - (R^2-OT^2) t.

Paul Yiu, Hyacinthos #21597

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