Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Ab, Ac = the orthogonal projections of A' on PB',PC', resp.

L1 = the radical axis of the circles (Ab, AbB') and (Ac, AcC')

M1 = the radical axis of the circle (Ab, AbC') and (Ac, AcB')

L1 and M1 are parallel (since both are perpendicular to AbAc)

Similarly L2, M2 and L3, M3

The triangles bounded by (L1,L2,L3) and (M1,M2,M3) are homothetic.

Homothetic center ? (in terms of the hom. coordinates of P)

Loci:

Which is the locus of P such that:

ABC, (L1,L2,L3) or (M1,M2,M3) are parallelogic?

A'B'C', (L1,L2,L3) or (M1,M2,M3) parallelogic?