EULER LINE

Let ABC be a triangle, P a point, A',B',C' the reflections of A,B,C in the perpendicular bisectors of BC, CA, AB, resp. and A"B"C" the circumcevian triangle of P 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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A ' = (a^2:-(b^2-c^2):b^2-c^2),

If P = (u:v:w), then

A'' = (((b^2-c^2)u+a^2v)((b^2-c^2)u-a^2w) : b^2(v+w)((b^2-c^2)u+a^2v) : -c^2(v+w)((b^2-c^2)u-a^2w))

etc.

A''B''C'' and ABC are perspective if and only if P lies on the Euler line. The perspector Q also lies on the Euler line.

If OP P PH = t : 1-t, then OQ : QH = (1+t) : -8t\cos A\cos B\cos C Here are some examples:

P Q

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G X(25)

O H

H X(24)

N X(3518)

L O

X(21) X(28)

X(22) G

X(23) X(468)

X(186 X(403)

Paul Yiu, Hyacinthos #12503