The triangle is called the primitive triangle with respect
to the triangle .
26.If one triangle be the polar triangle of another, the latter will be the polar triangle of the former.
Let be any triangle, the polar triangle: then
will be the polar triangle of .
For since is a pole of , the arc is a quadrant, and
since is a pole of , the arc is a quadrant (Art. 7);
therefore is a pole of (Art. 11). Also and are on
the same side of ; for and are by hypothesis on the
same side of , therefore is less than a quadrant; and
since is a pole of , and is less than a quadrant, and
are on the same side of .
Similarly it may be shewn that is a pole of , and that
and are on the same side of ; also that is a pole of
, and that and are on the same side of . Thus
is the polar triangle of .
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 , produced if necessary, meet the arcs ,
, produced if necessary, at the points and respectively;
then since is a pole of , the spherical angle is measured
by the arc (Art. 12). But and are each quadrants;
therefore and are together equal to a semicircle; that