PROPOSITION XXII 217. Theorem. If a tangent and a secant are drawn from a point to a circumference, the tangent is a mean proportional between the secant and its external segment. PROPOSITION XXIII 220. Theorem. If two chords intersect within a circle, the product of the segments of one equals the product of the segments of the other. Dem. X = Y D E B sim. Y х PROPOSITION XXIV 221. Theorem. The product of two sides of a triangle equals the product of the diameter of the circumscribed circle and the perpendicular to the third side from the opposite vertex. = 66 Dem. ABD = rt. 2 [insc. in semi 0] C=D [.“ same segment] sim. AB X AC = AD X AE 222. Cor. The diameter of a circle circumscribed about a triangle equals what? = PROPOSITION XXV 223. Theorem. The product of two sides of a triangle equals the product of the segments of the third side formed by the bisector of the opposite angle plus the square of the bisector. A B E Appl. Cons. Prove AB X AC = BD X CD + AD2 Dem. mut. = - Z X = X' (hyp.) B= E (insc. in same seg.)_ AD (AD + DE) AD + BD x CD ( prod. of seg. of AET EXERCISES 177. The sides of a triangle are 7, 8, and 12. The longest side of a similar triangle is 30. What are the other two sides ? 178. The bases of two similar triangles are 5 ft. 4 in. and 3 ft. 4 in. The altitude of the first is 4 ft. 10 in. What is the altitude of the second ? REGULAR POLYGONS — DEFINITION 224. A Regular Polygon is one that is both equiangular and equilateral. PROPOSITION XXVI 225. Theorem. Regular polygons of the same number of sides are similar. 179. In Prop. XIX, if AC, BC, 23, and 14 are respectively 48, 36, 27, and 25, what are 13 and AD ? 180. Two isosceles triangles are similar if any angle of one equals the homologous angle of the other. PROPOSITION XXVII 226. Theorem. A circle may be circumscribed about, or inscribed within, any regular polygon. Appl. ABCDE = reg. O. Circles may be circumscribed and inscribed. Cons. Take O, the centre of a O pass determined by ing through three consecutive I bisectors of ED vertices E, D, and C. and DC Join OB, OA, etc. OCD = isos. A 1. X = Y [°C A] (= radii) A ODE [ 2 s. and inc. _] OCB ОВ Dem. = = OE i.e. O, which passes through E, D, and C, also passes through B and (similarly] A, etc., and a circle is circumscribed about the regular polygon. 2. ED, DC, etc., are = distant chords of outer O, = by hyp. will touch sides of Lpolygon of] |