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.