circles

let ABC be the right triangle right angled at B.

let  O be the mid pt of the hypo. AC.

thru' O draw line parallel to BC meeting AB on P.

 so,

AP=PB..  conv of mid pt theorem

Angle OPA=OPB=90 each as line OP II BC and angle B is 90 degrees.

now consider triangles OAP and OBP,

we have

AP=PB

OP=OP.. common

angel OPA=OPB.. each 90 degrees.

 so the triangles are cong.... RHS rule.

OA=OB...CPCT

but OA=OC.. since O  is mid pt of AC

so we get

 OA=OB=OC

 Hence proved.