Theorem 19-4

If two distinct tangents are drawn to a circle from the same external point, the line segments from the external point to the points of tangency are equal in length; they form congruent right triangles when a line is drawn from the center of the circle to the external point.

The fact that the triangles described by Theorem 19-4 are congruent can beshown by the side-angle-side theorem from Chapter 17.

Figure 19.17 illustrates Theorem 19-4. Segments AB and AC are drawn tangent to the circle from an external point A. Segment AB is congruent to segment AC and angle BAO is congruent to angle CAO.

F IGURE 19.17 Equal length of tangents

Theorem 19-4 is used frequently in machine shop practice.

E XAMPLE 19.3: Theorem 19-4

In Figure 19.17, segments AB and AC are tangent to the circle. Given that the sum of the measures of all the angles in a triangle is 180 and that

Solution:

Corollary to Theorem 19-4

If two or more distinct pairs of tangents are drawn from points external to a circle, the bisectors of the angles formed by each pair of tangents will intersect at the center of the circle.

In Figure 19.18, the bisectors AB and A ′ B ′ of the angles formed by the tangents drawn from external points A and A ′ intersect at the center of the circle, O .

F IGURE 19.18 Intersection of bisectors at the center of the circle

Theorem 19-4 is the basis of the operation of a tool called a center head, which is used to find the center of the end faces of cylindrical bars or shafts.

F IGURE 19.18 Intersection of bisectors at the center of the circle

Theorems 19-5 and 19-6 are illustrated in Figure 19.19. Given that central angles AOD and BOD are congruent, it then follows that chord AD is congruent to chord BD , and arc AD is congruent to arc BD .

Theorem 19-5

In the same or congruent circles, congruent central angles subtend congruent

chords.

Theorem 19-6

In the same or congruent circles, congruent central angles subtend congruent

arcs.

Theorems 19-5 and 19-6 are illustrated in Figure 19.19. Given that central angles AOD and BOD are congruent, it then follows that chord AD is congruent to chord BD , and arc AD is congruent to arc BD .

F IGURE 19.19 Congruent central angles subtending congruent chords and arcs