Plane trigonometry, by S.L. Loney.

RADII OF THE ESCRIBED CIRCLES.

233

Produce AB and AC to L and M.
Bisect the angles CBL and
BCM by the lines BI, and CI1
and let these lines meet in I1. /
Draw 11D,, IEi, and IF / -
perpendicular to the three sides B
respectively. FB
The two triangles IDB and\!
IFB are equal in all respects, L/
so that I11F. \ 1JD
Similarly IE, = ID,. M
The three perpendiculars,1D1, 1iE1 and I1F1 being equal, the point 11 is the centre
of the required circle.
Now the area ABIC is equal to the sum of the
triangles ABC and I1BC; it is also equal to the sum of
the triangles ~1BA and I1CA.
Hence
A ABC + A I1BC = A 1CA + A IAAB..'. S +- ID. BC= -IE,. CA + -~IFx. AB,
i.e. S+ l. a= ~r. b +- -r. c.. b + c=- a = r + -c - +.
S = brlL 2 - =rl b - a = r, (s- a).
S.*o rl= —
s-a
Similarly it can be shewn that
S S
r2- and r3=S -b S-C