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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?

Cyclic Quads

Stage: 4 Challenge Level: Challenge Level:1

ABC is any triangle. D, E and F are arbitrary points on AC, AB and BC respectively. Circumcircles are drawn to the triangles ADE, CFD and BEF.

To change triangle ABC and points D, E and F, click on the 'Move' icon (top left) and then click and drag any of the points.

Follow the instructions to draw the circumcircle BEF in the dynamic diagram below.

What do you notice about the three circumcircles?

Can you prove your conjecture?


Created with GeoGebra


This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from together with a good help manual and Quickstart for beginners. You may be surprised at how easy it is to drawdynamic diagrams for yourself.