Find the midpoint of two points in barycentric coordinates.

Question

Though it is a very bad question for this site,I don't know the answer. So please help me.

What is the midpoints of the points $(a:b:c)$ and $(x:y:z)$ in barycentric coordinates?

Answer 1

You can convert to affine coordinates first. See Wikipedia Barycentric coordinate system. That is, $\, P = (a:b:c)/(a+b+c) + (x:y:z)/(x+y+z). \,$ Eliminating denominators we get $\, P = (2 a x + b x + c x + a y + a z : b x + a y + 2 b y + c y + b z : c x + c y + a z + b z + 2 c z). \,$ For example, the midpoint of $\, (a:0:0) \,$ and $\, (0:y:0) \,$ is $\, (ay:ay,0) = (1:1:0). \,$