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'Quads' printed from http://nrich.maths.org/
Created with GeoGebra

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.

NOTES AND BACKGROUND
By clicking on the link above you can go to the Geobgebra website
and download your own FREE, very easy to use,
educational mathematics software that combines dynamic geometry,
coordinate geometry, algebra and calculus. Also download the
Quickstart guide.
Even if you have never used dynamic geometry software you should be
able to draw this dynamic figure for yourself. You
could then ask some further "what if..." questions of your own and
change and add to the diagram to see what happens.
As we change the diagram some properties remain the same
(invariant) and others change. Some are invariant because of the
initial conditions of the problem but additional properties seem to
be invariant and we need to explain and prove why this is so.