In example 2 on pages 572, you were examining the locus of the centres of circles inscribed in a given angle. We will investigate the locus below.
In the sketch below are 4 circles inscribed in an angle. Look at the centres of the circles. Can you spot any relationship between the centres of the circles? The circles may be moved by dragging the centres. The angle can be changed by dragging the point on the upper arm. Does the relationship seem to hold when the angle is changed?
Now that you have an idea of what the locus of the centres of inscribed circles looks like, let's create it. In the sketch below is a single circle. Move the circle to get a picture in your mind of what the locus looks like. When you have an idea of what the locus looks like, trace the centre by clicking the button. Now, as the centre is dragged, the locus is traced out. What shape do we have? Is there anything special about this shape?
Take a closer look at the locus. How do you think it is related to the original angle? Look at the contact points of the circle. How do you think the triangles formed at the vertex are related? Play with the sketch below, clicking the buttons one at a time. Are you now able to determine how the locus is related to the original angle?
From the above sketches we were able to determine that the locus of a centre of a circle inscribed in an angle is the angle bisector. We can use this fact to help us create a circle inscribed in an angle. We know the following:
Using these we can create an inscribed circle in two ways:.
1. Given the centre (on the angle bisector)
Since the circle is tangent to each arm if a perpendicular is dropped from the centre to an arm, and the point of intersection is constructed, the circle will pass through that point .
2. Given the point of tangency (on arm of angle)
Use the fact that the circle will touch the arm at a point of tangency. The intersection of the angle bisector and the perpendicular line that goes through the point of tangency is the centre of the circle to be constructed.
The only question that remains is this: If we do this construction by hand (instead of with Sketchpad), how do we construct an angle bisector?
The sketches on this page were created with a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. Portions of this work were funded by the National Science Foundation (awards DMI 9561674 & 9623018).