. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . ri =- a (Art. 99). (1) Similarly it may be proved that if r2, r3 are the radii ofthe circles touching AC and AB respectively, S S n = 6 r* = - 105. To find the Distance be-tween the Centres of the Inscribedand Circumscribed Circles * of aTriangle. Let I and 0 be the incentreand circumcentre, respectively, ofthe triangle ABC, IA and ICbisect the angles BAC and BCA; * Often called the incentre and circumcen-tre of a 156 PLANE TRIGONOMETRY. therefore the arc BD is e
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . ri =- a (Art. 99). (1) Similarly it may be proved that if r2, r3 are the radii ofthe circles touching AC and AB respectively, S S n = 6 r* = - 105. To find the Distance be-tween the Centres of the Inscribedand Circumscribed Circles * of aTriangle. Let I and 0 be the incentreand circumcentre, respectively, ofthe triangle ABC, IA and ICbisect the angles BAC and BCA; * Often called the incentre and circumcen-tre of a 156 PLANE TRIGONOMETRY. therefore the arc BD is equal to the arc DC, and DOHbisects BC at right angles. Draw IM perpendicular to AC. Then Z DIC = A±S1 = BCD + BCI = DCL .-. DI = DC = 2Esin-. 2 Also, AI = IM cosec— = r cosec— * 2 2 .-. = 2Rr = EI-IF;that is, (E + 01) (E - 01) = 2 Br. .-. OI2 = E2-2Er. EXAMPLES. 1. The sides of a triangle are 18, 24, 30; find the radiiof its inscribed, escribed, and circumscribed circles. Ana. 6, 12, 18, 36, 15. 2. Prove that the area of the triangle ABC is 1 c2 2 cot A + cot B 3. Find the area of the triangle ABC when (1) a = 4, 6 = 10 ft., C = 30°. Arts. 10 sq. ft. (2) 6=5, c = 20 inches, A = 60°. sq. in. (3) a = 13, b = 14, c = 15 chains. 84 sq. chains. A T> 1111 4. Prove - = —-| 1 5. Prove r = g-g-in^Bsin^C cos^-A EXAMPLES. 157 6. Prove that the area of the triangle ABC is representedby each of the three expressions: 2 R2 sin A sin B sin C, rs, and Br (sin A + sin B + sin C). 7. If A = 60°, a = V3, 6 = V2, prove that the area
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Keywords: ., bookcentury1900, bookdecade1900, booksubjecttrigono, bookyear1902
