805 COROLLARY 3. In a tri-rectangular spherical triangle each side is a quadrant. 806 COROLLARY 4. Three planes passed through the center of a sphere, each perpendicular to the other two, divide the surface of the sphere into eight equal tri-rectangular triangles. SYMMETRICAL SPHERICAL POLYGONS 807 DEFINITION. Symmetrical spherical polygons are spherical polygons whose successive sides and angles are equal, each to each, but arranged in reverse order. Thus, the spherical triangles ABC and A'B'C' are mutually equilateral and equiangular, but their corresponding parts are arranged in reverse order. In general, two symmetrical spherical polygons cannot be made to coincide and are not equal. If, however, two symmetrical triangles are isosceles, they are equal. For if the symmetrical triangles. ABC and A'B'C' are isosceles, AC A'B', and AB = A'C'. ... the C triangles can be made to coincide. Hence the following A A' B 808 COROLLARY. Two symmetrical isosceles spherical tri angles are equal. PROPOSITION XVI. THEOREM 809 Two triangles on the same sphere or equal spheres are equal, 1 If two sides and the included angle of the one are equal respectively to two sides and the included angle of the other; 2 If two angles and the included side of the one are equal respectively to two angles and the included side of the other; 3 If the three sides of the one are equal respectively to the three sides of the other; Provided that the equal parts in each case are arranged in the same order. PROOF In cases 1 and 2, the A, like plane A, can be made to coincide, and are therefore equal. §§ 162, 167 In case 3, the face angles of the corresponding triedral angles at the center of the sphere are respectively equal. § 241 ... the corresponding diedral / are equal. § 593 .. the of the spherical ▲ are respectively equal. § 789 Case 1 Q. E. D. PROPOSITION XVII. THEOREM 810 Two triangles on the same sphere or equal spheres are symmetrical, 1 If two sides and the included angle of the one are equal respectively to two sides and the included angle of the other; 2 If two angles and the included side of the one are equal respectively to two angles and the included side of the other; 3 If the three sides of the one are equal respectively to the three sides of the other; Provided that the equal parts in each case are arranged in reverse order. In each case, A and B are the given ▲. But ▲ C and B are symmetrical by const. § 809 ... the ▲ A, which is equal to the ▲ C, is symmetrical with the Δ Β. Q. E. D. PROPOSITION XVIII. THEOREM 811 If two triangles on the same sphere or equal spheres are mutually equiangular, they are mutually equilateral, and are either equal or symmetrical. HYPOTHESIS. A and A' are mutually equiangular spherical triangles. CONCLUSION. AA and A' are mutually equilateral, and are either equal or symmetrical. PROOF Let P and P' be the respective polar ▲ of A and A'. Since AA and A' are mutually equiangular, A P and P' are mutually equilateral. § 800 Therefore AA and A' are equal if their equal parts are arranged in the same order (Figs. 1 and 2), § 809, Case 3 § 810, Case 3 Q. E. D. and they are symmetrical if their equal parts are arranged in reverse order (Figs. 1 and 3). EXERCISES 1267 What spherical triangle is its own polar triangle? 1268 A spherical surface can be passed through four points in the same plane if the four points are concyclic. How many solutions? PROPOSITION XIX. THEOREM 812 Two symmetrical spherical triangles are equiva lent. A B C B HYPOTHESIS. ABC and A'B'C' are symmetrical spherical A. PROOF Let P and P' be the poles of the small circles passing through A, B, C, and A', B', C', respectively. The arcs AB, AC, and BC are equal to the arcs A'B', A'C', and B'C' respectively. § 807 •. the chords of the arcs AB, AC, and BC are equal to the chords of the arcs A'B', A'C', and B'C' respectively. § 243 .. the plane ▲ formed by these chords are equal. § 172 § 299 Draw the great circle arcs PA, PB, PC, and P'A', P'B', P'C'. These great circle arcs are all equal. Hence the ▲ PAB and P'A'B' are isosceles and symmetrical. § 764 § 810, Case 3 8 808 Adding, ▲ ABC ≈ A'B'C'. Ax. 1 Should the poles fall without the triangles, one of the equations must be subtracted from the sum of the other two. Q. E. D. |