Re: [EMHL] Circle of diametre [U;M] where U is the circumcenter
- Dear Michel
I suppose that coordinates or equations of all objects in triangle geometry
were written as soon as barycentrics or trilinears were found, more than
150 years ago!
As for your circles, it is easy to get equation of any circle <Gamma> in the
triangle ABC plane.
Name p(A), p(B), p(C) the respective powers of the vertices A, B, C wrt
circle <Gamma>, then an equation of <Gamma> in barycentrics is:
a²yz + b²zx + c²xy - (x + y + z)(p(A)x + p(B)y + p(C)z) = 0
You get the coordinates of its center as the pole of the line at infinity.
Notice that the line with equation:
p(A)x + p(B)y + p(C)z = 0
is the radical axis of <Gamma> and the ABC-circumcircle.
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