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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.