ABCD is a convex quadrilateral with diagonals AC and BD intersecting at X. The circumcircles of the triangles AXB, BXC, CXD and DXA have centres P, Q, R and S respectively.

Move the vertices of the quadrilateral ABCD.

What do you notice about the quadrilateral PQRS? 

Make a conjecture about PQRS and prove your conjecture.

X

A = (-0.9991018888888913, 3.65617411111111)
B = (1.153016659259257, 3.653571266666665)
C = (1.1872874444444426, 1.4958132222222205)
D = (-0.9470450000000024, 1.4958132222222205)
poly1, a, b, c, d = Polygon(A, B, C, D)
e = Segment(A, C)
f = Segment(B, D)
O = Intersect(e, f)
g = Circle(A, O, B)
h = Circle(C, O, B)
k = Circle(A, O, D)
p = Circle(O, C, D)
i = LineBisector(A, B)
j = LineBisector(O, B)
P = Intersect(i, j)
l = LineBisector(B, C)
m = LineBisector(O, C)
Q = Intersect(l, m)
n = LineBisector(D, C)
q = LineBisector(O, C)
R = Intersect(n, q)
r = LineBisector(A, D)
s = LineBisector(A, O)
S = Intersect(r, s)
poly2, f_1, g_1, h_1, i_1 = Polygon(P, Q, R, S)