[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point
Denote:
A', B', C' = the reflections of P in BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
M1, M2, M3 = the reflections of Ma, Mb, Mc in PA', PB', PC', resp.
Which is the locus of P such that the circumcircles of APM1, BPM2, CPM3 are coaxial?
[Angel Montesdeoca]:
*** The circumcircles of APM1, BPM2, CPM3 are coaxial if P lies on a circum-septic passing through points X(1), X(3), X(4) and X(5), of barycentric equation:
-2 a^8 c^2 x^6 y+3 a^6 b^2 c^2 x^6 y+3 a^4 b^4 c^2 x^6 y-7 a^2 b^6 c^2 x^6 y+3 b^8 c^2 x^6 y+8 a^6 c^4 x^6 y-9 a^4 b^2 c^4 x^6 y-a^2 b^4 c^4 x^6 y+2 b^6 c^4 x^6 y-12 a^4 c^6 x^6 y+9 a^2 b^2 c^6 x^6 y+8 a^2 c^8 x^6 y-3 b^2 c^8 x^6 y-2 c^10 x^6 y-3 a^8 c^2 x^5 y^2+7 a^6 b^2 c^2 x^5 y^2-3 a^4 b^4 c^2 x^5 y^2-3 a^2 b^6 c^2 x^5 y^2+2 b^8 c^2 x^5 y^2+4 a^6 c^4 x^5 y^2-8 a^4 b^2 c^4 x^5 y^2+9 a^2 b^4 c^4 x^5 y^2-5 b^6 c^4 x^5 y^2+6 a^4 c^6 x^5 y^2-3 a^2 b^2 c^6 x^5 y^2-6 b^4 c^6 x^5 y^2-12 a^2 c^8 x^5 y^2+4 b^2 c^8 x^5 y^2+5 c^10 x^5 y^2+4 a^8 c^2 x^4 y^3-6 a^6 b^2 c^2 x^4 y^3-6 a^4 b^4 c^2 x^4 y^3+14 a^2 b^6 c^2 x^4 y^3-6 b^8 c^2 x^4 y^3-13 a^6 c^4 x^4 y^3+18 a^4 b^2 c^4 x^4 y^3-7 a^2 b^4 c^4 x^4 y^3+2 b^6 c^4 x^4 y^3+9 a^4 c^6 x^4 y^3-18 a^2 b^2 c^6 x^4 y^3+9 b^4 c^6 x^4 y^3+5 a^2 c^8 x^4 y^3-5 c^10 x^4 y^3+6 a^8 c^2 x^3 y^4-14 a^6 b^2 c^2 x^3 y^4+6 a^4 b^4 c^2 x^3 y^4+6 a^2 b^6 c^2 x^3 y^4-4 b^8 c^2 x^3 y^4-2 a^6 c^4 x^3 y^4+7 a^4 b^2 c^4 x^3 y^4-18 a^2 b^4 c^4 x^3 y^4+13 b^6 c^4 x^3 y^4-9 a^4 c^6 x^3 y^4+18 a^2 b^2 c^6 x^3 y^4-9 b^4 c^6 x^3 y^4-5 b^2 c^8 x^3 y^4+5 c^10 x^3 y^4-2 a^8 c^2 x^2 y^5+3 a^6 b^2 c^2 x^2 y^5+3 a^4 b^4 c^2 x^2 y^5-7 a^2 b^6 c^2 x^2 y^5+3 b^8 c^2 x^2 y^5+5 a^6 c^4 x^2 y^5-9 a^4 b^2 c^4 x^2 y^5+8 a^2 b^4 c^4 x^2 y^5-4 b^6 c^4 x^2 y^5+6 a^4 c^6 x^2 y^5+3 a^2 b^2 c^6 x^2 y^5-6 b^4 c^6 x^2 y^5-4 a^2 c^8 x^2 y^5+12 b^2 c^8 x^2 y^5-5 c^10 x^2 y^5-3 a^8 c^2 x y^6+7 a^6 b^2 c^2 x y^6-3 a^4 b^4 c^2 x y^6-3 a^2 b^6 c^2 x y^6+2 b^8 c^2 x y^6-2 a^6 c^4 x y^6+a^4 b^2 c^4 x y^6+9 a^2 b^4 c^4 x y^6-8 b^6 c^4 x y^6-9 a^2 b^2 c^6 x y^6+12 b^4 c^6 x y^6+3 a^2 c^8 x y^6-8 b^2 c^8 x y^6+2 c^10 x y^6+2 a^8 b^2 x^6 z-8 a^6 b^4 x^6 z+12 a^4 b^6 x^6 z-8 a^2 b^8 x^6 z+2 b^10 x^6 z-3 a^6 b^2 c^2 x^6 z+9 a^4 b^4 c^2 x^6 z-9 a^2 b^6 c^2 x^6 z+3 b^8 c^2 x^6 z-3 a^4 b^2 c^4 x^6 z+a^2 b^4 c^4 x^6 z+7 a^2 b^2 c^6 x^6 z-2 b^4 c^6 x^6 z-3 b^2 c^8 x^6 z-9 a^4 b^4 c^2 x^5 y z+18 a^2 b^6 c^2 x^5 y z-9 b^8 c^2 x^5 y z+9 a^4 b^2 c^4 x^5 y z-3 b^6 c^4 x^5 y z-18 a^2 b^2 c^6 x^5 y z+3 b^4 c^6 x^5 y z+9 b^2 c^8 x^5 y z-2 a^10 x^4 y^2 z+4 a^8 b^2 x^4 y^2 z+4 a^6 b^4 x^4 y^2 z-16 a^4 b^6 x^4 y^2 z+14 a^2 b^8 x^4 y^2 z-4 b^10 x^4 y^2 z-15 a^6 b^2 c^2 x^4 y^2 z+27 a^4 b^4 c^2 x^4 y^2 z-9 a^2 b^6 c^2 x^4 y^2 z-3 b^8 c^2 x^4 y^2 z+12 a^6 c^4 x^4 y^2 z+14 a^4 b^2 c^4 x^4 y^2 z-20 a^2 b^4 c^4 x^4 y^2 z+12 b^6 c^4 x^4 y^2 z-16 a^4 c^6 x^4 y^2 z+3 a^2 b^2 c^6 x^4 y^2 z+b^4 c^6 x^4 y^2 z+6 a^2 c^8 x^4 y^2 z-6 b^2 c^8 x^4 y^2 z-9 a^8 c^2 x^3 y^3 z+18 a^6 b^2 c^2 x^3 y^3 z-18 a^2 b^6 c^2 x^3 y^3 z+9 b^8 c^2 x^3 y^3 z+15 a^6 c^4 x^3 y^3 z-33 a^4 b^2 c^4 x^3 y^3 z+33 a^2 b^4 c^4 x^3 y^3 z-15 b^6 c^4 x^3 y^3 z-3 a^4 c^6 x^3 y^3 z+3 b^4 c^6 x^3 y^3 z-3 a^2 c^8 x^3 y^3 z+3 b^2 c^8 x^3 y^3 z+4 a^10 x^2 y^4 z-14 a^8 b^2 x^2 y^4 z+16 a^6 b^4 x^2 y^4 z-4 a^4 b^6 x^2 y^4 z-4 a^2 b^8 x^2 y^4 z+2 b^10 x^2 y^4 z+3 a^8 c^2 x^2 y^4 z+9 a^6 b^2 c^2 x^2 y^4 z-27 a^4 b^4 c^2 x^2 y^4 z+15 a^2 b^6 c^2 x^2 y^4 z-12 a^6 c^4 x^2 y^4 z+20 a^4 b^2 c^4 x^2 y^4 z-14 a^2 b^4 c^4 x^2 y^4 z-12 b^6 c^4 x^2 y^4 z-a^4 c^6 x^2 y^4 z-3 a^2 b^2 c^6 x^2 y^4 z+16 b^4 c^6 x^2 y^4 z+6 a^2 c^8 x^2 y^4 z-6 b^2 c^8 x^2 y^4 z+9 a^8 c^2 x y^5 z-18 a^6 b^2 c^2 x y^5 z+9 a^4 b^4 c^2 x y^5 z+3 a^6 c^4 x y^5 z-9 a^2 b^4 c^4 x y^5 z-3 a^4 c^6 x y^5 z+18 a^2 b^2 c^6 x y^5 z-9 a^2 c^8 x y^5 z-2 a^10 y^6 z+8 a^8 b^2 y^6 z-12 a^6 b^4 y^6 z+8 a^4 b^6 y^6 z-2 a^2 b^8 y^6 z-3 a^8 c^2 y^6 z+9 a^6 b^2 c^2 y^6 z-9 a^4 b^4 c^2 y^6 z+3 a^2 b^6 c^2 y^6 z-a^4 b^2 c^4 y^6 z+3 a^2 b^4 c^4 y^6 z+2 a^4 c^6 y^6 z-7 a^2 b^2 c^6 y^6 z+3 a^2 c^8 y^6 z+3 a^8 b^2 x^5 z^2-4 a^6 b^4 x^5 z^2-6 a^4 b^6 x^5 z^2+12 a^2 b^8 x^5 z^2-5 b^10 x^5 z^2-7 a^6 b^2 c^2 x^5 z^2+8 a^4 b^4 c^2 x^5 z^2+3 a^2 b^6 c^2 x^5 z^2-4 b^8 c^2 x^5 z^2+3 a^4 b^2 c^4 x^5 z^2-9 a^2 b^4 c^4 x^5 z^2+6 b^6 c^4 x^5 z^2+3 a^2 b^2 c^6 x^5 z^2+5 b^4 c^6 x^5 z^2-2 b^2 c^8 x^5 z^2+2 a^10 x^4 y z^2-12 a^6 b^4 x^4 y z^2+16 a^4 b^6 x^4 y z^2-6 a^2 b^8 x^4 y z^2-4 a^8 c^2 x^4 y z^2+15 a^6 b^2 c^2 x^4 y z^2-14 a^4 b^4 c^2 x^4 y z^2-3 a^2 b^6 c^2 x^4 y z^2+6 b^8 c^2 x^4 y z^2-4 a^6 c^4 x^4 y z^2-27 a^4 b^2 c^4 x^4 y z^2+20 a^2 b^4 c^4 x^4 y z^2-b^6 c^4 x^4 y z^2+16 a^4 c^6 x^4 y z^2+9 a^2 b^2 c^6 x^4 y z^2-12 b^4 c^6 x^4 y z^2-14 a^2 c^8 x^4 y z^2+3 b^2 c^8 x^4 y z^2+4 c^10 x^4 y z^2-9 a^8 b^2 x^3 y^2 z^2+20 a^6 b^4 x^3 y^2 z^2-6 a^4 b^6 x^3 y^2 z^2-12 a^2 b^8 x^3 y^2 z^2+7 b^10 x^3 y^2 z^2+9 a^8 c^2 x^3 y^2 z^2-28 a^4 b^4 c^2 x^3 y^2 z^2+24 a^2 b^6 c^2 x^3 y^2 z^2-5 b^8 c^2 x^3 y^2 z^2-20 a^6 c^4 x^3 y^2 z^2+28 a^4 b^2 c^4 x^3 y^2 z^2-20 b^6 c^4 x^3 y^2 z^2+6 a^4 c^6 x^3 y^2 z^2-24 a^2 b^2 c^6 x^3 y^2 z^2+20 b^4 c^6 x^3 y^2 z^2+12 a^2 c^8 x^3 y^2 z^2+5 b^2 c^8 x^3 y^2 z^2-7 c^10 x^3 y^2 z^2-7 a^10 x^2 y^3 z^2+12 a^8 b^2 x^2 y^3 z^2+6 a^6 b^4 x^2 y^3 z^2-20 a^4 b^6 x^2 y^3 z^2+9 a^2 b^8 x^2 y^3 z^2+5 a^8 c^2 x^2 y^3 z^2-24 a^6 b^2 c^2 x^2 y^3 z^2+28 a^4 b^4 c^2 x^2 y^3 z^2-9 b^8 c^2 x^2 y^3 z^2+20 a^6 c^4 x^2 y^3 z^2-28 a^2 b^4 c^4 x^2 y^3 z^2+20 b^6 c^4 x^2 y^3 z^2-20 a^4 c^6 x^2 y^3 z^2+24 a^2 b^2 c^6 x^2 y^3 z^2-6 b^4 c^6 x^2 y^3 z^2-5 a^2 c^8 x^2 y^3 z^2-12 b^2 c^8 x^2 y^3 z^2+7 c^10 x^2 y^3 z^2+6 a^8 b^2 x y^4 z^2-16 a^6 b^4 x y^4 z^2+12 a^4 b^6 x y^4 z^2-2 b^10 x y^4 z^2-6 a^8 c^2 x y^4 z^2+3 a^6 b^2 c^2 x y^4 z^2+14 a^4 b^4 c^2 x y^4 z^2-15 a^2 b^6 c^2 x y^4 z^2+4 b^8 c^2 x y^4 z^2+a^6 c^4 x y^4 z^2-20 a^4 b^2 c^4 x y^4 z^2+27 a^2 b^4 c^4 x y^4 z^2+4 b^6 c^4 x y^4 z^2+12 a^4 c^6 x y^4 z^2-9 a^2 b^2 c^6 x y^4 z^2-16 b^4 c^6 x y^4 z^2-3 a^2 c^8 x y^4 z^2+14 b^2 c^8 x y^4 z^2-4 c^10 x y^4 z^2+5 a^10 y^5 z^2-12 a^8 b^2 y^5 z^2+6 a^6 b^4 y^5 z^2+4 a^4 b^6 y^5 z^2-3 a^2 b^8 y^5 z^2+4 a^8 c^2 y^5 z^2-3 a^6 b^2 c^2 y^5 z^2-8 a^4 b^4 c^2 y^5 z^2+7 a^2 b^6 c^2 y^5 z^2-6 a^6 c^4 y^5 z^2+9 a^4 b^2 c^4 y^5 z^2-3 a^2 b^4 c^4 y^5 z^2-5 a^4 c^6 y^5 z^2-3 a^2 b^2 c^6 y^5 z^2+2 a^2 c^8 y^5 z^2-4 a^8 b^2 x^4 z^3+13 a^6 b^4 x^4 z^3-9 a^4 b^6 x^4 z^3-5 a^2 b^8 x^4 z^3+5 b^10 x^4 z^3+6 a^6 b^2 c^2 x^4 z^3-18 a^4 b^4 c^2 x^4 z^3+18 a^2 b^6 c^2 x^4 z^3+6 a^4 b^2 c^4 x^4 z^3+7 a^2 b^4 c^4 x^4 z^3-9 b^6 c^4 x^4 z^3-14 a^2 b^2 c^6 x^4 z^3-2 b^4 c^6 x^4 z^3+6 b^2 c^8 x^4 z^3+9 a^8 b^2 x^3 y z^3-15 a^6 b^4 x^3 y z^3+3 a^4 b^6 x^3 y z^3+3 a^2 b^8 x^3 y z^3-18 a^6 b^2 c^2 x^3 y z^3+33 a^4 b^4 c^2 x^3 y z^3-3 b^8 c^2 x^3 y z^3-33 a^2 b^4 c^4 x^3 y z^3-3 b^6 c^4 x^3 y z^3+18 a^2 b^2 c^6 x^3 y z^3+15 b^4 c^6 x^3 y z^3-9 b^2 c^8 x^3 y z^3+7 a^10 x^2 y^2 z^3-5 a^8 b^2 x^2 y^2 z^3-20 a^6 b^4 x^2 y^2 z^3+20 a^4 b^6 x^2 y^2 z^3+5 a^2 b^8 x^2 y^2 z^3-7 b^10 x^2 y^2 z^3-12 a^8 c^2 x^2 y^2 z^3+24 a^6 b^2 c^2 x^2 y^2 z^3-24 a^2 b^6 c^2 x^2 y^2 z^3+12 b^8 c^2 x^2 y^2 z^3-6 a^6 c^4 x^2 y^2 z^3-28 a^4 b^2 c^4 x^2 y^2 z^3+28 a^2 b^4 c^4 x^2 y^2 z^3+6 b^6 c^4 x^2 y^2 z^3+20 a^4 c^6 x^2 y^2 z^3-20 b^4 c^6 x^2 y^2 z^3-9 a^2 c^8 x^2 y^2 z^3+9 b^2 c^8 x^2 y^2 z^3-3 a^8 b^2 x y^3 z^3-3 a^6 b^4 x y^3 z^3+15 a^4 b^6 x y^3 z^3-9 a^2 b^8 x y^3 z^3+3 a^8 c^2 x y^3 z^3-33 a^4 b^4 c^2 x y^3 z^3+18 a^2 b^6 c^2 x y^3 z^3+3 a^6 c^4 x y^3 z^3+33 a^4 b^2 c^4 x y^3 z^3-15 a^4 c^6 x y^3 z^3-18 a^2 b^2 c^6 x y^3 z^3+9 a^2 c^8 x y^3 z^3-5 a^10 y^4 z^3+5 a^8 b^2 y^4 z^3+9 a^6 b^4 y^4 z^3-13 a^4 b^6 y^4 z^3+4 a^2 b^8 y^4 z^3-18 a^6 b^2 c^2 y^4 z^3+18 a^4 b^4 c^2 y^4 z^3-6 a^2 b^6 c^2 y^4 z^3+9 a^6 c^4 y^4 z^3-7 a^4 b^2 c^4 y^4 z^3-6 a^2 b^4 c^4 y^4 z^3+2 a^4 c^6 y^4 z^3+14 a^2 b^2 c^6 y^4 z^3-6 a^2 c^8 y^4 z^3-6 a^8 b^2 x^3 z^4+2 a^6 b^4 x^3 z^4+9 a^4 b^6 x^3 z^4-5 b^10 x^3 z^4+14 a^6 b^2 c^2 x^3 z^4-7 a^4 b^4 c^2 x^3 z^4-18 a^2 b^6 c^2 x^3 z^4+5 b^8 c^2 x^3 z^4-6 a^4 b^2 c^4 x^3 z^4+18 a^2 b^4 c^4 x^3 z^4+9 b^6 c^4 x^3 z^4-6 a^2 b^2 c^6 x^3 z^4-13 b^4 c^6 x^3 z^4+4 b^2 c^8 x^3 z^4-4 a^10 x^2 y z^4-3 a^8 b^2 x^2 y z^4+12 a^6 b^4 x^2 y z^4+a^4 b^6 x^2 y z^4-6 a^2 b^8 x^2 y z^4+14 a^8 c^2 x^2 y z^4-9 a^6 b^2 c^2 x^2 y z^4-20 a^4 b^4 c^2 x^2 y z^4+3 a^2 b^6 c^2 x^2 y z^4+6 b^8 c^2 x^2 y z^4-16 a^6 c^4 x^2 y z^4+27 a^4 b^2 c^4 x^2 y z^4+14 a^2 b^4 c^4 x^2 y z^4-16 b^6 c^4 x^2 y z^4+4 a^4 c^6 x^2 y z^4-15 a^2 b^2 c^6 x^2 y z^4+12 b^4 c^6 x^2 y z^4+4 a^2 c^8 x^2 y z^4-2 c^10 x^2 y z^4+6 a^8 b^2 x y^2 z^4-a^6 b^4 x y^2 z^4-12 a^4 b^6 x y^2 z^4+3 a^2 b^8 x y^2 z^4+4 b^10 x y^2 z^4-6 a^8 c^2 x y^2 z^4-3 a^6 b^2 c^2 x y^2 z^4+20 a^4 b^4 c^2 x y^2 z^4+9 a^2 b^6 c^2 x y^2 z^4-14 b^8 c^2 x y^2 z^4+16 a^6 c^4 x y^2 z^4-14 a^4 b^2 c^4 x y^2 z^4-27 a^2 b^4 c^4 x y^2 z^4+16 b^6 c^4 x y^2 z^4-12 a^4 c^6 x y^2 z^4+15 a^2 b^2 c^6 x y^2 z^4-4 b^4 c^6 x y^2 z^4-4 b^2 c^8 x y^2 z^4+2 c^10 x y^2 z^4+5 a^10 y^3 z^4-9 a^6 b^4 y^3 z^4-2 a^4 b^6 y^3 z^4+6 a^2 b^8 y^3 z^4-5 a^8 c^2 y^3 z^4+18 a^6 b^2 c^2 y^3 z^4+7 a^4 b^4 c^2 y^3 z^4-14 a^2 b^6 c^2 y^3 z^4-9 a^6 c^4 y^3 z^4-18 a^4 b^2 c^4 y^3 z^4+6 a^2 b^4 c^4 y^3 z^4+13 a^4 c^6 y^3 z^4+6 a^2 b^2 c^6 y^3 z^4-4 a^2 c^8 y^3 z^4+2 a^8 b^2 x^2 z^5-5 a^6 b^4 x^2 z^5-6 a^4 b^6 x^2 z^5+4 a^2 b^8 x^2 z^5+5 b^10 x^2 z^5-3 a^6 b^2 c^2 x^2 z^5+9 a^4 b^4 c^2 x^2 z^5-3 a^2 b^6 c^2 x^2 z^5-12 b^8 c^2 x^2 z^5-3 a^4 b^2 c^4 x^2 z^5-8 a^2 b^4 c^4 x^2 z^5+6 b^6 c^4 x^2 z^5+7 a^2 b^2 c^6 x^2 z^5+4 b^4 c^6 x^2 z^5-3 b^2 c^8 x^2 z^5-9 a^8 b^2 x y z^5-3 a^6 b^4 x y z^5+3 a^4 b^6 x y z^5+9 a^2 b^8 x y z^5+18 a^6 b^2 c^2 x y z^5-18 a^2 b^6 c^2 x y z^5-9 a^4 b^2 c^4 x y z^5+9 a^2 b^4 c^4 x y z^5-5 a^10 y^2 z^5-4 a^8 b^2 y^2 z^5+6 a^6 b^4 y^2 z^5+5 a^4 b^6 y^2 z^5-2 a^2 b^8 y^2 z^5+12 a^8 c^2 y^2 z^5+3 a^6 b^2 c^2 y^2 z^5-9 a^4 b^4 c^2 y^2 z^5+3 a^2 b^6 c^2 y^2 z^5-6 a^6 c^4 y^2 z^5+8 a^4 b^2 c^4 y^2 z^5+3 a^2 b^4 c^4 y^2 z^5-4 a^4 c^6 y^2 z^5-7 a^2 b^2 c^6 y^2 z^5+3 a^2 c^8 y^2 z^5+3 a^8 b^2 x z^6+2 a^6 b^4 x z^6-3 a^2 b^8 x z^6-2 b^10 x z^6-7 a^6 b^2 c^2 x z^6-a^4 b^4 c^2 x z^6+9 a^2 b^6 c^2 x z^6+8 b^8 c^2 x z^6+3 a^4 b^2 c^4 x z^6-9 a^2 b^4 c^4 x z^6-12 b^6 c^4 x z^6+3 a^2 b^2 c^6 x z^6+8 b^4 c^6 x z^6-2 b^2 c^8 x z^6+2 a^10 y z^6+3 a^8 b^2 y z^6-2 a^4 b^6 y z^6-3 a^2 b^8 y z^6-8 a^8 c^2 y z^6-9 a^6 b^2 c^2 y z^6+a^4 b^4 c^2 y z^6+7 a^2 b^6 c^2 y z^6+12 a^6 c^4 y z^6+9 a^4 b^2 c^4 y z^6-3 a^2 b^4 c^4 y z^6-8 a^4 c^6 y z^6-3 a^2 b^2 c^6 y z^6+2 a^2 c^8 y z^6=0.
Pairs {P, Q}, where Q is the second point of intersection of the circumcircles of PAM1, PBM2, PCM3:
{X(1), X(3579) = midpoint of X(3) and X(40)}, {X(5), X(140)= midpoint of X(3) and X(5)}.
If P=X(3),
Q =(a^2 (a^6-a^4 (3 b^2+c^2)+a^2 (3 b^4+b^2 c^2-c^4)-(b^2-c^2)^3)
(2 a^14
-a^12 (9 b^2+13 c^2)
+2 a^10 (7 b^4+19 b^2 c^2+18 c^4)
-a^8 (5 b^6+42 b^4 c^2+56 b^2 c^4+55 c^6)
+a^6 (-10 b^8+31 b^6 c^2+28 b^4 c^4+27 b^2 c^6+50 c^8)
+a^4 (13 b^10-26 b^8 c^2-b^6 c^4-5 b^4 c^6+10 b^2 c^8-27 c^10)
-a^2 (b^2-c^2)^2 (6 b^8-3 b^6 c^2-4 b^4 c^4-3 b^2 c^6-8 c^8)
+(b^2-c^2)^5 (b^2+c^2)^2 ) : ... : ....),
with (6-9-13)-search numbers (6.15594320395733, 6.30024330748564, -3.56224697894753),
and lies on lines {186,1291}, {550,1263}.
If P=X(4) then P,A and M1 are collinear, etc...
Angel Montesdeoca
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