PROPOSITION XX. THEOREM

813 In an isosceles spherical triangle, the angles opposite the equal sides are equal; conversely, if two angles of a spherical triangle are equal, the sides opposite are equal, and the triangle is isosceles.

Let the arc AD bisect the Z A.

Then ▲ ADB and ADC are symmetrical.

.. Z B = Z C.

CONVERSELY

HYPOTHESIS. In the spherical ▲ ABC, B = ≤ C (Fig. 2).

§ 810, Case 1

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PROPOSITION XXI. THEOREM

814 If two angles of a spherical triangle are unequal, the sides opposite are unequal, and the greater side is opposite the greater angle; conversely, if two sides of a spherical triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side.

HYPOTHESIS. In the spherical ▲ ABC, B > < C.

CONCLUSION. AC > AB.

PROOF

A

D

Draw the great circle arc BD, making ≤ DBC

Then DB = DC.

Since AD + DB > AB,

AD + DC > AB; or AC > AB.

CONVERSELY

HYPOTHESIS. In the spherical ▲ ABC, AC > AB.

B

Z C.

§ 813

§ 793

Since both these conclusions are contrary to the hypothesis,

<B> < C.

Q. E. D.

THEOREMS

1269 Two vertical spherical angles are equal.

1270 If a spherical triangle is equilateral, it is also equiangular. 1271 If a spherical triangle is equiangular, it is also equilateral. 1272 Parallel circles of a sphere have the same poles.

1273 If one of two polar triangles is isosceles, the other is also isosceles.

1274 The arcs bisecting the base angles of an isosceles spherical triangle are equal.

1275 If two sides of a spherical triangle are quadrants, the third side measures the opposite angle.

1276 The polar triangle of a bi-rectangular triangle is bi-rectangular. 1277 A tri-rectangular triangle and its polar are identical.

1278 The three perpendicular bisectors of the sides of a spherical triangle meet in a point.

1279 The three medians of a spherical triangle meet in a point.

1280 The three bisectors of the angles of a spherical triangle meet in a point.

1281 The intersection of two spherical surfaces is the circumference of a circle.

1282 Two symmetrical spherical polygons are equivalent.

1283 An exterior angle of a spherical triangle is less than the sum of the two opposite interior angles.

1284 The sum of the angles of a spherical pentagon is greater than six right angles.

1285 A radius of a sphere perpendicular to the plane of a circle of the sphere passes through the center of the circle.

1286 Each side of a spherical triangle is greater than the difference of the other two sides.

1287 If a right spherical triangle has one side greater than a quadrant, it has a second side greater than a quadrant.

1288 If one of two great circles passes through the pole of the other, their circumferences intersect at right angles.

PROBLEMS OF CONSTRUCTION

1289 To bisect an arc of a great circle.

1290 To bisect a spherical angle.

1291 To inscribe a circle in a spherical triangle.

1292 To circumscribe a circle about a spherical triangle.

1293 At a given point in a great circle to construct a spherical angle equal to a given spherical angle.

1294 To construct a spherical triangle having given two sides and the included angle.

1295 To construct a spherical triangle having given two angles and the included side.

1296 To construct a spherical triangle having given the three sides.

1297 To construct a spherical triangle having given the three angles. 1298 To draw the circumference of a circle through any three points on the surface of a sphere.

1299 Pass a plane through a given point within a sphere so that the section shall be the least circle; the largest circle.

1300 Pass a spherical surface through four given points.

1301 Through a given point on the surface of a sphere pass a plane tangent to the sphere.

1302 Through a given straight line without a sphere pass a plane tangent to the sphere.

1303 To draw a great circle through a given point on a sphere tangent to a given small circle of the sphere.

1304 Pass a spherical surface through two given points and having its center in a given line.

With a given radius to construct a spherical surface :

1305 Passing through three given points.

1306 Tangent to two given spheres and passing through a given point.

1307 Tangent to three given spheres.

1308 Tangent to three given planes.

PROBLEMS IN LOCI

1309 Find the locus of a point at a given distance from a fixed point. 1310 Find the locus of a line tangent to a given sphere at a given point in the surface of the sphere.

1311 Find the locus of a line drawn from a given point without a given sphere tangent to the sphere.

1312 Find the locus of a point at a given distance from the surface of a given sphere.

1313 Find the locus of the center of a given sphere whose surface passes through a fixed point.

1314 Find the locus of a point on the surface of a given sphere equidistant from three fixed points on the surface of the sphere.

1315 Find the locus of a point on the surface of a given sphere equidistant from two fixed points on the surface of the sphere.

1316 Find the locus of the center of a given sphere whose surface is tangent to a fixed line.

1317 Find the locus of the center of a sphere whose surface passes through the vertices of a given triangle.

1318 Find the locus of the center of a sphere whose surface passes through two given points.

1319 Find the locus of the center of a given sphere tangent to two given spheres.

1320 Find the locus of the center of a sphere tangent to three given planes.

1321 Find the locus of the center of a given sphere tangent to two given intersecting planes.

1322 Find the locus of a sphere tangent to the lateral faces of a given regular prism.

1323 Find the locus of a sphere tangent to the lateral faces of a given regular pyramid.

1324 Find the locus of the middle point of a straight line drawn from a fixed point without to the surface of a given sphere.

1325 Find the locus of the center of a section of a given sphere whose plane passes through a given straight line without the sphere.