Plane and solid geometry . s a quad- rant. 3. .*. C is the pole of AB, 4. Likewise b is the pole of AC, and A is the pole of BC. 5. ..A ABC is the polar of A ab^C, Reasons 1. § 912. 2. § 912. 3. § 914. 4. By steps simi- lar to 1- 3. 5. § 943. 946. Historical Note. The properties of polar triangles were dis-covered about 1626 by Albert Girard, a Dutch mathematiciau, bornin Lorraine about 1595. They were also discovered independently andabout the same time by Snell,, an infant prodigy, who at the age oftwelve was familiar with the standard mathematical works of that timeand who is re
Plane and solid geometry . s a quad- rant. 3. .*. C is the pole of AB, 4. Likewise b is the pole of AC, and A is the pole of BC. 5. ..A ABC is the polar of A ab^C, Reasons 1. § 912. 2. § 912. 3. § 914. 4. By steps simi- lar to 1- 3. 5. § 943. 946. Historical Note. The properties of polar triangles were dis-covered about 1626 by Albert Girard, a Dutch mathematiciau, bornin Lorraine about 1595. They were also discovered independently andabout the same time by Snell,, an infant prodigy, who at the age oftwelve was familiar with the standard mathematical works of that timeand who is remembered as the discoverer of the well-known law ofrefraction of light. Ex. 1519. Determine the polar triangle of a spherical triangle havingtwo of its sides quadrants and the third side equal to 70°; 110°; (90 — a)° ;(90 4- ay. 434 SOLID GEOMETRY Propositiox XII. Theorem 947. In two polar triangles each angle of one andthat side of tlve other of which its vertex is tlie pole aretogether equal, numerically, to ISO. E a Given polar A ABC and abc, with sides denoted by a, 6, c,and a, V, c\ respectively. To prove: (a) Z^+a = 180°, Z5+?> = 180^ Zc+c=180°;[h) Z^+a=180°, Z ^+6=180°, Z (7+c=180°. (<z) Argument Only 1. Let arcs AB and AC (prolonged if necessary) intersect arc5c at D and E, respectively; then C^B = 90° and EB^ = 90°. 2. .-. ci) +^5= 180°. 3. .-. CE + ED + ED-^- BB = 180°; EB-\-o! = 180°. 4. But EB is the measure of Z . .•.Z^ + a = 180°. 6. Likewise ^V = 180°, and Z C + c = 180°. (5) The proof of (6) is left as an exercise for the Let BG prolonged meet A^B^ at H and A^C^ at K, 948. Question. In the history of mathematics, why are polar tri-angles frequently spoken of as supplemental triangles ? Ex. 1520. The ani^^les of a spherical triangle are 75°, 85^, and 146°Find the sides of its polar triangle. Ex. 1521. If a spherical triangle is equilateral, its polar triangle isequiangular; and conversely. BOOK IX 4
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