Here are the highlights:

View attachment 39594
Your equation, in terms of x, y, and z is

\(\displaystyle | \vec{x} + \vec{y} |^2 z + | \vec{y} + \vec{z} |^2 x - y^2 | \vec{x} - \vec{z} | = | \vec{x} - \vec{z} | xz\)

Writing this out:

\(\displaystyle (x^2 + z^2 + 2xz~cos( \theta ) ) z + (y^2 + z^2 + 2yz~cos( \pi - \theta ) )x - y^2(x + z) = (x + z)xz\)

(where \(\displaystyle \theta\) is the angle ADC.)

Expanding these out and noting that \(\displaystyle cos( \pi - \theta ) = - cos( \theta )\) the y terms drop out, leaving

\(\displaystyle x^2z + xz^2 = (x + z)xz\), which is true for all x, z.

-Dan