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 IV.

59

equal each to each; hence AH = AB. Again the As
AHI, ABI have the sides AH, AI and the included Z
in the one equal to the two sides AB, AI and the included Z in the other;.'. the L HIA = BIA.
Now, bisect the Z ABI by the line BO, and it is
evident, by letting fall Ls on the four sides of the
quadrilateral ABIH from the point 0, that the four iLs
are equal to one another. Hence the 0, having 0 as
centre, and any of these i-s as radius, will be inscribed
in the quadrilateral;.*. HI is a tangent to the escribed
0, which touches AB externally. In like manner, it
may be proved that HI touches the escribed 0, which
touches AC externally. Hence HI is the fourth common tangent to these two circles.
c D L B P I
Cor. 1.-If D be the middle point of the base BC,
the 0, whose centre is D and whose radius is DG, is
orthogonal to the two escribed Os which touch BC
produced.
For, let P be the point of contact of the escribed 0,
which touches AB externally, then
PD = CP - CD = b (a + b + o) - 5a= ~(b + c);
and since BH is bisected in G, and BC in D,
DG = IOH = (AB + AC) = l (b + o);
therefore the 0, whose centre is D and radius DG, will
cut orthogonally the 0 which touches at P.