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.