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 VI.

155

Again, HN = HP = QL, and NS = ER = EL;.-. HS = EQ;.. (see 6, Prop. 1, Section V.) the tangents at E and H to the Os
A
E
~M~~~~~ C
ES and HQ meet on their 0 of similitude;.'. C is a point on the
O of similitude of the Os ES and IQ; and therefore these Os
subtend equal L s at C. Also, three common tangents of the Os HQ,
ES, PNR, viz., QL, SN, KF, are concurrent;.-. (see Ex. 48,
Section II., Book III.) C must be the point of concurrence of
three other common tangents to the same Os. Hence the second
transverse common tangent to HQ and ES must pass through C;
and since C is a point on their ( of similitude, this transverse
common tangent must bisect the L ACB. In like manner it is
proved that the bisectors of A and B are transverse common tangents to the Os ES and GT, and to HQ and GT, respectively.
Hence, we have the following elegant construction:-Let V be
the point of concurrence of the three bisectors of the L s of the A
ABC. In the A s VAB, VBC, VCA, describe three Os: these Os
will evidently, taken in pairs, have VB, VC, VA as transverse
common tangents; then to the same pairs of Os draw the three
other transverse tangents; these will be respectively ED, GF,
HI; and the Os described touching the triads of lines AB, AC,
ED; AB, BC, GF; AB, BC, HI, will be the required circles.
This construction is due to Steiner, and the foregoing simple
and elementary proof to Dr. Hart (see Quarterly Journal, vol. i.
p. 219).
62. If a transversal passing through a fixed point 0 cut any
number of fixed lines in the points A, B, C, &c., and if P be a
point such that
1 1 1 1
OP OA OB O 4

the locus of P is a right line.