ANSWER FOR ANGLE PROPERTY 1 (ANGLE AT CENTRE)

Angle at Centre = 2 x Angle at Circumference
 
Note: As long as Angle(AOB) is not more than 180°, the ratio Angle(AOB)/Angle(AXB) will always be equal to 2. If Angle(AOB) is more than 180°, the ratio will still be equal to 2 but you can only see this if you open the GSP sketch IT3EMCirclesAngleCentre.gsp using Geometer's Sketchpad as JavaSketchpad cannot support it. The sketch also shows the angles being shaded which JavaSketchpad cannot support.
 
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