The distance from the incenter to an acute vertex of a right triangle I'm seeking an alternative proof of this result:

Given $\triangle ABC$ with right angle at $A$. Point $I$ is the intersection of the three angle lines. (That is, $I$ is the incenter of $\triangle ABC$.) Prove that 
  $$|CI|^2=\frac12\left(\left(\;|BC|-|AB|\;\right)^2+|AC|^2\right)$$

My Proof. Draw $ID \perp AB$, $IE\perp BC$, and $IF\perp CE$.
We have $|ID|=|IE|=|IF|=x$. Since $\triangle ADI$ is right isosceles triangle, we also have that $|AD|=|ID|=x$. Respectively, we have: $$|ID|=|IF|=|IE|=|AD|=|AF|=x$$ 
$\triangle BDI=\triangle BEI \Rightarrow |BD|=|BE|=y$. And $|CE|=|CF|=z$
We have:
$$|CI|^2=|CE|^2+|IE|^2=x^2+z^2 \tag{1}$$
And 
$$\begin{align}\frac12\left(\left(|BC|-|AB|\right)^2+|AC|^2\right) &=\frac12\left(\left(\;\left(y+z\right)-\left(x+y\right)\;\right)^2+\left(x+z\right)^2\right) \\[4pt]
&=\frac12\left(\left(x-z\right)^2+\left(x+z\right)^2\right) \\[4pt]
&=\frac22\left(x^2+z^2\right) \\[4pt]
&=x^2+z^2
\tag{2}\end{align}$$
From $(1);(2)$ we are done. $\square$


Let the circle about $B$ through $C$ meet the extension of $\overline{AB}$ at the point $C^\prime$. By symmetry and a little angle chasing in isosceles $\triangle CBC^\prime$, we find that $\triangle CIC^\prime$ is an isosceles right triangle. Consequently,
$$|IC|^2 + |IC|^2 = |IC|^2 + |IC^\prime|^2 = |CC^\prime|^2 = |AC|^2 + |AC^\prime|^2 = |AC|^2 + ( |BC| - |AB| )^2$$
