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.

MISCELLANEOUS EXERCISES. 23

235

29. If the area of the triangle BIB2B33, Ex. 27, be given, the
locus of each point is a circle.
flem.-Here we have theB
ratios BiB2: S3B2, and B2B3:Si B2 given. Hence the ratio
of the rectangle BjB2. B2B3
sin BiB2B3: S3B2. SiB2. sin L
BiB2B3 is given: but the for.
mer rectangle is given; thereforethe rectangle S3B2. S11B2
sin 1B1B2B3 is given; now,
the angles B1B32S3, SjB2B3 are
given. Hence the angle B1
B13B3 ~ S3B2S1 is given. Let
its value be denoted by a; S3
therefore BiB2B3 = a + S31
]33S1 Hence, taking th
upper sign, the problem is reduced to the following. The
base S3S1 of a triangle S31B2S1
is given in magnitude and position, and the rectangle 53132. SiB2
sin (a - 53132S1) is given in magnitude, tofind the locus of B2, which
is solved as follows:-Upon S3S, describe a segment of a circle
S3LS, EEuc. III., xxxiii.] containing an angle S3LS1 equal to a.
Join SiL; then the angle B2SiL is equal to a - S3B2Si. Hence,
by hypothesis, S3B. SIB2. sin B2S1L is given; hut SIB2. sin
B2SjL is equal to LB2 sin a; therefore the rectangle S31B2. LB2,
is given. Hence the locus of B2 is a circle.
30. If Q, Q' he the inverses of the Brocard points of a triangle,
with respect to its circumeirele, the pedal triangles of Q, Q' are10, equal to one another; 20, the sides of one are perpendicular
to the corresponding sides of the other; 3', e~ch is inversely
similar to the original. (M'CAY.)
31. If the area of the triangle formed by three corresponding
lines of three figures directly similar he given, the envelopes of
its sides are circles whose centres are the invariable points of the
three figures.
32. If the area of the polygon formed by n corresponding lines
of nfigures directly similar described on the sides of a harmonic
polygon of n sides be given, the envelopes of the sides are circles
whose centres are the in-variable points of the harmonic polygon.

33. The four symmedian lines of a harmonic octagon form a
harmonic pencil,