<Polygon-2025-276>, hint

 Let O be the circumcenter of ABCDE. Let OA be the vector

OH(1)=OA+OB+OC, OM(1)=1/2(OD+OE), Suppose H(1)M(1) and H(2)M(2) meet at  P

OP=(1-x)(OA+OB+OC)+x 1/2 (OD+OE). Similarly OH(2)=OB+OC+OD  OM(2)=1/2(OE+OA)

(1-x)(OA+OB+OC)+x1/2(OD+OE)=(1-y)(OB+OC+OD)+y 1/2(OE+OA) Finally x=y=2/3

Hence OP=1/3(OA+OB+OC+OD+OE)