The Inscribed Right Triangle Theorem can be used to justify why this construction works.
Consider a radius of
Kriz is learning a graphic program. By default, the program shows segment and circle The segment's endpoint can be moved anywhere outside of While endpoint can be moved anywhere. An eye-like shape appears on the screen when is tangent to the circle. Give it a try!
Kriz can't quite place point in position to see the eye-like shape appear. Help Kriz out!
How can a tangent line from a point outside of the given circle be constructed?
Since point is a point outside should be the point of tangency in order for to be tangent to the circle. On the example shape, by extending it can be observed that is the point of tangency.
Constructing a tangent from an outer point will help locate the point of tangency for a tangent drawn from Recall the steps in constructing a tangent.
In this case, point is the outer point through which the tangent line is drawn. To get the example shape, move point A to the left as shown and then follow the steps.
As can be seen, the points where the circles intersect are the points of tangency. Therefore, point should be on these points.
If and are tangent segments to then
Consider two triangles.
These two triangles can be visualized in the diagram.
Combining all of this information, it can be said that the hypotenuse and one leg of are congruent to the hypotenuse and the corresponding leg of
The points and are the points where the segments touch the circle. If and find
From the graph, it can be seen that and are tangent segments with a common endpoint outside By the External Tangent Congruence Theorem, and are congruent.