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?
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.
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?
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
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.