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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