The words inscriptible, circumscriptible, escriptible mean capable of being inscribed in, circumscribed about, escribed to, a circle.

315. Prove from th. 16 a and th. 11 that the sum of the

interior angles of any triangle equals a straight angle.

DEFINITIONS.

Section 5. Two Circles.

Two circles are said to touch or to be tangent when their circumferences have one, and only one, point in common.

They are said to be internally or externally tangent according as one circle lies within or without the other. The more accurate expression, a tangent circumference, is often used instead of a tangent circle.

The line determined by the centers of two circles is called their center-line; the segment of the center-line, between the centers, is called their center-segment.

If two circles have a common center they are said to be concentric.

The expression concentric circumferences is also used.

EXERCISES.

319. A triangle is inscribed in a circle. Prove that the sum of three angles, one in each segment of the circle, exterior to the triangle, equals a perigon.

320. A perpendicular from the orthocenter of a triangle to a side, produced to the circumference of the circumscribed circle, is bisected by that side.

321. The bisectors of any angle of an inscribed quadrilateral and the opposite exterior angle meet on the circumference.

322. If the diagonals of an inscribed quadrilateral bisect each other, what kind of a quadrilateral is it?

323. If two consecutive sides of a convex hexagon inscribed in a circle are respectively parallel to their opposite sides, the remaining sides are parallel to each other.

324. The bisectors of the angles formed by producing the opposite sides of an inscribed quadrilateral to meet, are perpendicular to each other. (A proof may be based on cors. 2 and 1 of th. 14.)

325. If the diagonals of an inscribed quadrilateral are perpendicular to each other, the line through their intersection perpendicular to any side bisects the opposite side. (Brahmagupta's theorem.)