Page:Spherical Trigonometry (1914).djvu/30

12 Two triangles, $$ABC, A''BC$$, which have a side $$BC$$ common, and whose other sides belong to the same great circles, are called colunar triangles, as they together make up a lune. $$A''$$ is the point diametrically opposite to $$A$$ on the sphere.

If $$A, B, C''$$ be diametrically opposite to $$A$$, $$B$$, $$C$$ respectively, the triangle $$ABC$$ has three colunar triangles, namely, $$A''BC$$, $$BCA$$, and $$CAB$$.

Antipodal triangles are triangles whose respective vertices are diametrically opposite to one another in pairs; such, for example, are the triangles $$ABC$$, $$ABC''$$.

25. Polar triangle. Let $$ABC$$ be any spherical triangle, and let the points $$A'$$, $$B'$$, $$C'$$ be those poles of the arcs $$BC$$, $$CA$$, $$AB$$ respectively which lie on the same sides of them as the opposite angles $$A$$, $$B$$, $$C$$; then the triangle $$A'B'C'$$ is said to be the polar triangle of the triangle $$ABC$$.



Since there are two poles for each side of a spherical triangle, eight triangles can be formed having for their angular points poles of the sides of the given triangle; but there is only one triangle in which these poles $$A'$$, $$B'$$, $$C'$$ lie towards the same parts with the corresponding angles $$A$$, $$B$$, $$C$$; and this is the triangle which is known under the name of the polar triangle.