A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

BOOK FOURTH.
SECTION I.
ADDITIONAL PROPOSITIONS.
Prop. 1. —If a circle be inscribed in a triangle, the
distances from the angular points of the triangle to the
points of contact on the sides are respectively equal to the
remainders that are left, when the lengths of the sides are
taken separately from their half sum.
Dem.-Let ABC be the A, A
D, E, F, the points of contact.
Now, since the tangents from an
external point are equal, we have
AE = AF, BD = BF, CD = CE.
Hence AE + BC = AB + CE
= half sum of the three sides \
BC, CA, AB; and denoting these
sides by the letters a, b, c, respectively, and half their sum B D
by s, we have
AE + a = s
therefore AE = s - a.
In like manner BD = - b; CE = s -.
Cor. 1.-If r denote the radius of the inscribed 0,
the area of the triangle = rs.
For, let 0 be the centre of the inscribed 0, then we
have
BC. r = 2 BO,
CA. r =2 A COA,
AB. r=2 AAOB;
therefore (BC + CA + AB) r = 2 A ABC;