Plane and solid geometry . K= (b + 2a (Fig. 1), or - q2 -I- 52 _ e^2a a^j^b-c^ b- a- + b 2 a J\ 2 a j 2ab -\-a^-hb^-c^ 2 ab-a^-b^-{-c 2a 2a ^(a+b -^c)(a-^b—c)(c-ha— b)(c—a-\-b) 4a^ ^ , _ l{a-\-b + c)(a-^b — c)(c-{-a — b)(c — a-{-b) 6. Now let a + 6 + c = 2 s. Then a + b-c = 2s — 2c = 2(s-c)]a-b-{-c = 2s-2b = 2(s-b),b + c — a = 2s — 2a = 2(s — a). 2. § 447. 3. § 456. 4. § 309. 5. § 54, 13. 6. § 54, 3. BOOK IV 221 Argument 7. Then j.^^^J^TW^a) ^s-b) 2(s-c)^ 4. or 8. a 2 = Vs(s — a)(s — 6)(s — c). Keasons 7. § 309. 8. § 309. Ex. 833. Find the area of a triangle whose sides are 7, 10, and 1
Plane and solid geometry . K= (b + 2a (Fig. 1), or - q2 -I- 52 _ e^2a a^j^b-c^ b- a- + b 2 a J\ 2 a j 2ab -\-a^-hb^-c^ 2 ab-a^-b^-{-c 2a 2a ^(a+b -^c)(a-^b—c)(c-ha— b)(c—a-\-b) 4a^ ^ , _ l{a-\-b + c)(a-^b — c)(c-{-a — b)(c — a-{-b) 6. Now let a + 6 + c = 2 s. Then a + b-c = 2s — 2c = 2(s-c)]a-b-{-c = 2s-2b = 2(s-b),b + c — a = 2s — 2a = 2(s — a). 2. § 447. 3. § 456. 4. § 309. 5. § 54, 13. 6. § 54, 3. BOOK IV 221 Argument 7. Then j.^^^J^TW^a) ^s-b) 2(s-c)^ 4. or 8. a 2 = Vs(s — a)(s — 6)(s — c). Keasons 7. § 309. 8. § 309. Ex. 833. Find the area of a triangle whose sides are 7, 10, and 13. Ex. 834. If the sides of a triangle are a, 6, and c, write the formulafor the altitude upon h ; upon c. (See Prop. IV, Arg. 7.) Ex. 835. In triangle ABC, a = 8, 6 = 12, c = 16 ; find the area oftriangle ABC; the altitude upon b ; the altitude upon c. Proposition V. Theorem 491. TJte area of a triangle is equal to one half theproduct of its perimeter and the radius of the A b Given A ABC, with area T, sides a, by and c, and radius ofinscribed circle r. To prove r= ^ {a + b + c)T, Outline of Proop 1. Area of A OBC=:\ a • r; area of A OCA = i- 6 . r; areaoi AOAB = \ c-r. 2. ., T^^{a + h + c)r. 492. Cor. The area of any polygon circumscribedabout a circle is equal to one half its perimeter multi-plied by the radius of the inscribed circle. 224 PLANE GEOMETFwY
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