The Art and Craft of Problem Solving

266 CHAPTER 8 GEOMETRY FOR AMERICANS

8.2. 17 Inscribed Right Triangles. Let AB be the diameter of a circle, and let C be an

arbitrary point on the circle. Then LBCA = 90°. Conversely, if AB is a diameter and

LBCA = 90°, then C must lie on the circle. (See left figure below.)

A I!------"TB

c

B

8.2.18 Cyclic Quadrilaterals. The quadrilateral ABCD is cyclic if its vertices lie on a

circle. The points A, B, C, D are called concyclic. (See right figure above.)

(a) A quadrilateral is cyclic if and only if a pair of opposite angles are supplementary.

(b) Points A, B, C, Dare concyclic if and only if LACB = LADB.

With cyclic quadrilaterals you automatically get circles and pairs of equal angles "for

free." This extra structure often provides useful information. Get in the habit of locat­

ing (or creating) cyclic quadrilaterals.

Circles and Triangles

One can spend a lifetime exploring the interplay between circles and triangles. First

we shall look at the inscribed and circumscribed circles of a triangle.

In the diagram below, triangle ABC is inscribed in circle n, since each vertex of

the triangle lies on this circle. We call n the circumscribed circle or circumcircle

of triangle ABC. The center of n, or circumcenter, is 0, and the length OA is the

circumradius.

Likewise, circle 12 is inscribed in triangle ABC because it is tangent to each side of

the triangle. We call it the inscribed circle or incircle. The center of 12, the incenter,

is I, one point of tangency is D, and thus the length of I D is the inradius.