<Qua-2025-769>, hint

Let O be the circumcenter  of ABCD and vectors OA,OB,OC,OD,OM be a,b,c,d,m.

H(1)=a+b+m, H(2)=b+c+m, H(3)=c+d+m, H(4)=d+a+m,

H(1)H(2)=a-c H(3)H(4)=c-a, H(2)H(3)=b-d, H(1)H(4)=b-d

E=1/2(a+b), F=1/2(c+d).

H(1)H(3)=a+b-c-d, EF=1/2(a+b-c-d)