Connect AO.

Because o is the midpoint of BC.

So ao =1/2bc = bo = co.

AO=BO (S)

∠OAN=∠B=45 (A)

BM=AN (S)

According to SAS, △OBM equals △OAN.

So MO = no

∠BOM=∠AON

Because BOM+MOA = 90.

So ∠AON+∠MOA=∠MON=90.

Swimming foam = none

So △OMN is an isosceles right triangle.