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.

13OOK Iii.

PAC, are = two right Z s (xxii.);.'. L PFD = PAC;
and since PFAE is a quadri- E
lateral in a circle, the z EAP p A
= EFP;.-. PFD + PFE = PAC
+ PAE = two right Zs. Hence
the points D, F, E, are collinear. / /
Cor. 1.-If the feet of the
-s drawn from any point P
to the sides of the A ABC be
collinear, the locus of P is the
O described about the triangle.
Cor. 2.-If four lines be given, a point can be found
such, that the feet of the B
four I s from it on the lines
willbe collinear. Forlet the
four lines be AB, AC, DB,
DF. These lines form four A
As. Let the Os described
about two of the As-say
AFE, ODE-intersect inP; D
then it is evident that the
feet of the ILs from P on
the four lines will be collinear.
Cor. 3.-The Os described about the As ABC, DBF,
each passes through the point P. This follows because
the feet of the -Ls from P on the sides of these As are
collinear.
Prop. 13.-If the perpendiculars of a triangle be produced to meet the circumference of
the circumscribed circle, the parts of
the perpendiculars intercepted between their point of intersection and \
the circumference are bisected by the
sides of the triangle.
Let AD, CF intersect in 0; pro- F /
duce CF to meet the 0 in G; then
OF = FG.
Dem. —TheZ AOF = COD (I. xv.) and AFO = CDO,
D2