Plane trigonometry, by S.L. Loney.

HEIGHTS AND DISTANCES.

219

If A be the point at which the greatest angle is
subtended then A must be the point where a circle drawn
through P and Q touches the road.
[For, take any other point A' on AB and join it to P
cutting the circle in B' and join A'Q and B'Q.
Then Z PA'Q < z PB'Q (Euc. I. 16),
and therefore < Z PAQ (Euc. III. 21).]
Let the angle QAB be called 0. Then (Euc. III. 32)
the angle A PQ is 0 also.
Hence 180~= sum of the angles of the triangle PAB
=0 +(a+ 0)+ 3,
so that 0 =90- -
From the triangles PAQ and QAB we have
PQ sinc an AQ _ sin_ _ sin 3
AQ sin 0' c sin A QB sin( + a)'
Hence, by multiplication, we have
PQ sin a sin /
c sin 0 sin (0 + a)'
sin a sin 8
-cots2- cos — _2
a+/3 aS.. PQ= c sin a sin 3 sec -- sec 2
2 2