Angles at Circumference & Centre (Proof)

Case 1: BC is Diameter

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change their positions and the length of the arc AB

Instructions:

1. Adjust the position and the length of the arc AB.
2. Check all boxes.
3. Repeat steps 1 & 2 for other positions and lengths of the arc AB.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41520


Case 2: BC is not Diameter & O is inside Segment BAC

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change their positions and the length of the arc AB
• shift the point C to change its position

Instructions:

1. Adjust the position and the length of the arc AB.
2. Adjust the position of the point C.
3. Check all boxes.
4. Repeat steps 1 ~ 3 for other positions and lengths of the arc AB and the point C.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41521


Case 3: BC is not Diameter & O is outside Segment BAC

Below shows a circle, with centre O.
The points A, B, C and X are points on the circumference of the circle.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change their positions and the length of the arc AB
• shift the point C to change its position

Instructions:

1. Adjust the position and the length of the arc AB.
2. Adjust the position of the point C.
3. Check all boxes.
4. Repeat steps 1 ~ 3 for other positions and lengths of the arc AB and the point C.

Note: To see the animation, check all boxes, then click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/41522