In right triangle ACM, from Pythagorean theorem,

AM^2=AC^2+CM^2

In the right triangle BMN, from the pythagorean theorem,

MN^2=BM^2-BN^2

In right triangle AMN, from Pythagorean theorem,

AN^2=AM^2-MN^2

=(AC^2+CM^2)-(BM^2-BN^2)

=AC^2+CM^2-BM^2+BN^2

Because AM is the center line on the side of BC

So CM=BM

So an 2 = AC 2+BN 2

That is, a square -BN square =AC square.