The proof in my post “A Computer Algebra Aided Proof in Plane Geometry” can be significantly simplified if complex numbers are used.
Consider complex numbers in FIg. 1.
Fig. 1
Since
we have,
i.e.,
Similarly,
and
give
Let denote the slope of line connecting
and
respectively.
From
we see that
lie on the same line.
Notice that since
.
See also “Treasure Hunt with Complex Numbers“.