20. If a, b, p denote the sides of a right-angled triangle about the right angle, and the perpendicular from the right angle on the hypotenuse, + = . 21. If, upon the greater segment AB of a line AC, divided in extreme and mean ratio, an equilateral triangle ABD be described, and CD joined, CD2 = 2AB2. 22. If a variable line, whose extremities rest on the circumferences of two given concentric circles, subtend a right angle at any fixed point, the locus of its middle point is a circle. BOOK III. THEORY OF THE CIRCLE ________________ DEFINITIONS. i. Equal circles are those whose radii are equal. This is a theorem, and not a definition. For if two circles have equal radii, they are evidently congruent figures, and therefore equal. From this way of proving this theorem Props. xxvi.–xxix. follow as immediate inferences. ii. A chord of a circle is the line joining two points in its circumference. If the chord be produced both ways, the whole line is called a secant, and each of the parts into which a secant divides the circumference is called an arc—the greater the major conjugate arc, and the lesser the minor conjugate arc.—Newcomb. iii. A right line is said to touch a circle when it meets the circle, and, being produced both ways, does not cut it; the line is called a tangent to the circle, and the point where it touches it the point of contact. In Modern Geometry a curve is considered as made up of an infinite number of points, which are placed in order along the curve, and then the secant through two consecutive points is a tangent. Euclid’s definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections); and even for these the modern definition is better. iv. Circles are said to touch one another when they meet, but do not intersect. There are two species of contact:—1. When each circle is external to the other.2. When one is inside the other. The following is the modern definition of curve-contact:— When two curves have two, three, four, &c., consecutive points common, they have contact of the first, second, third, &c., orders. v. A segment of a circle is a figure bounded by a chord and one of the arcs into which it divides the circumference. vi. Chords are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal. vii. The angle contained by two lines, drawn from any point in the circumference of a segment to the extremities of its chord, is called an angle in the segment. viii. The angle of a segment is the angle contained between its chord and the tangent at either extremity. A theorem is tacitly assumed in this Definition, namely, that the angles which the chord makes with the tangent at its extremities are equal. We shall prove this further on. ix. An angle in a segment is said to stand on its conjugate arc.