given: triangle with an angle C with adjacent sides a and b and opposite side c.
bisect b (or a) at right angle with line passing thru opposite angle
works easier with obtuse angles such that intersection is internal to triangle - otherwise you end up hanging on the outside of the triangle which feels awkward.
we choose to bisect a side other than c so we don't bisect angle C
label this new line x and the divided parts of b as y and z
from basic Pythagorean we can state the following:
x2 + z2 = c2
x2 + y2 = a2
we can eliminate x2 easily
isolating c2
removing one of the parts of b (since y + z = b then z = b - y)
expanding (b - y)2
remove the y^2s
from SOH-CAH-TOA we can state that
cos(C) = y/a
y = a * cos(C)
substitute for y