Circle of Apollonius for an Angle $v$

1.

Open Algebra View and Graphics View under  View in  Menu.

2.

Select the Segment  tool and draw a line segment in Graphics View.

3.

Select the Perpendicular Bisector  tool to draw the perpendicular bisector of the line segment.

4.

Select the Point on Object  tool and draw a point on your line segment. The point will usually get the name C.

5.

Select the Circle with Center through Point  tool, and click C, then A. You should get a circle with the arc $AB$, which is the segment from $A$ that extends counterclockwise toward $B$.

6.

Select the Angle  tool and click the points A, C, and B, in that order. This angle is called the central angle $\alpha$, which is subtended by the arc $AB$.

7.

Adjust the position of point $C$ so that angle $\alpha$ is twice as big as $v$—that is, $\alpha =2v$.

8.

Select the Point on Object  tool and draw a point on the circle which is not on the arc $AB$. The point will usually get the name D.

9.

Select the Angle  tool and click the points A, D, and B, in that order. This angle is called an inscribed angle, which is subtended by the arc $AB$.

Adjust point D. As long as D doesn’t cross A or B, you’ll see that every inscribed angle subtended by the arc $AB$ will always have the same size as $v$! You’ve just drawn the Circle of Apollonius for the angle $v=\frac{\alpha }{2}$.