Page:Spherical Trigonometry (1914).djvu/31

§27] The triangle $$ABC$$ is called the primitive triangle with respect to the triangle $$A'B'C'$$.

26. If one triangle be the polar triangle of another, the latter will be the polar triangle of the former.

Let $$ABC$$ be any triangle, $$A'B'C'$$ the polar triangle: then $$ABC$$ will be the polar triangle of $$A'B'C'$$.



For since $$B'$$ is a pole of $$AC$$, the arc $$AB'$$ is a quadrant, and since $$C'$$ is a pole of $$BA$$, the arc $$AC'$$ is a quadrant (Art. 7); therefore $$A$$ is a pole of $$B'C'$$ (Art. 11). Also $$A$$ and $$A'$$ are on the same side of $$B'C'$$; for $$A$$ and $$A'$$ are by hypothesis on the same side of $$BC$$, therefore $$A'A$$ is less than a quadrant; and since $$A$$ is a pole of $$B'C'$$, and $$AA'$$ is less than a quadrant, $$A$$ and $$A'$$ are on the same side of $$B'C'$$.

Similarly it may be shewn that $$B$$ is a pole of $$C'A'$$, and that $$B$$ and $$B'$$ are on the same side of $$C'A'$$; also that $$C$$ is a pole of $$A'B'$$, and that $$C$$ and $$C'$$ are on the same side of $$A'B'$$. Thus $$ABC$$ is the polar triangle of $$A'B'C'$$.

27. The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.

For let the arc $$B'C'$$, produced if necessary, meet the arcs $$AB$$, $$AC$$, produced if necessary, at the points $$D$$ and $$E$$ respectively; then since $$A$$ is a pole of $$B'C'$$, the spherical angle $$A$$ is measured by the arc $$DE$$ (Art. 12). But $$B'E$$ and $$C'D$$ are each quadrants; therefore $$DE$$ and $$B'C'$$ are together equal to a semicircle; that