Wentworths Solid Geometry

Cover Wentworths Solid Geometry
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This line will cut the arcs DCF and GCE in separate points D and E ; and if we revolve the line A D about A as a fixed point until D coincides with C, we shall have a line from A to C equal to the line AD. In like manner, we can draw a line from J5 to C equal to the line BE. Therefore a line can be drawn from A to B through C that is equal to the sum of the lines AD and BE, and hence is less than the line A DEB by the line DE. Therefore no line which does not pass through C can be the shortest ...line from ^ to j5. Therefore the shortest line from A to B passes through C. But C is any point in the arc AB. Therefore the shortest line from A to B passes through every point of the arc AB, and consequently coincides with the arc AB. Therefore the shortest line from A to B is the great-circle arc ^5. Q. E. D. EXERCISE 106 1. The three medians of a spherical triangle are concurrent. 2. To construct with a given radius a spherical surface that passes through three given points. 3. To construct with a given radius a spherical surface that passes through two given points and is tangent to a given plane.

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Wentworths Solid Geometry
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