<Solid-2025-257>, hint

Let O be the center of circumscribed sphere of ABCD,

and let vectors OA=a, OB=b, OC=c, OD=d.

P=t(a+c)/2+(1-t)(b+d)/2, X=xa+(1-x)b, Y=yc+(1-y)d, P=kX+(1-k)Y. 

Then we get followings 2kx=t, 2k-2kx=1-t, 2y-2ky=t, 2(1-k-y+ky)=1-t

Finally  we get x=y=t, 2k=1.