sliding secant lines

Arthur Lee
This applet shows how to find the size of a pair of arcs cut by two secant lines through a point inside or on a circle. The main purpose is the explain a basic property of angle and arc in circle geometry from a transformational perspective, thus without relying on angle chasing and formal proofs. Drag the slider slowly to slide the line BP, then AP, to the centre of the circle. While the lines are sliding, note how the total size of both red arcs is not changed. This can be explained by the intuitive idea that parallel chords should cut equal arcs on both ends. It is intuitive due to symmetry of a circle (imagine a diameter perpendicular to that pair of parallel chords as a line of symmetry). When the secant lines are moved to the centre, the size of the red arcs will be easily found from the angle at centre.
In this figure, when you finish sliding the secant lines, you will see the arcs and , which should be equal. Their combined size should be same as that of the original arcs AB and CD. The point P can also be put on a circle, overlapping with C and D.
You may also click the checkbox to cut the circle into equal parts for estimating the size of the arcs.
Another applet is made for exploration only.