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.

194

194 ~A SEQUEL TO EUCLID.

CoR. 2.-.-.The lines' CP,, CP2, CP, are concurrent, and
the locus of their point of concurrence is the circle of similitude.
The substance of this Section is taken from ]J~athesis, vol. ii.,
page 73. Propositions 1-6 are due to M. G. TARRty, and Proposition 6 to NEIJBERG.
Exercises.
1. If in the invariable triangle be inscribed triangles equiangular to the triangle of similitude, so that the vertex corresponding to SI will be on the side P2P3, &c., the centre of similitude
of the inscribed triangles is the director point.
2. If V1, V2, Y3 he the centres of the circles which are the
loci of the points C1, C 2, C3; then the sum of the angles PI, S 1, V1
is equal to the sum of P2, S2, V2, equal to the sum Of P3, S3, VSb,
equal to two right angles.
S. The system of multiples for which the director point is the
mean centre of the in-variable points is 0, cosec al, (t2 cosec a2,
03 cosec a3.
4. The director point, and eitber the triangle of similitude or
the invariahie triangle suffice to determine the figures FI, F2, F3.
5. Prove that the triangles SIS2S3', S2S3S1', S3S1Sz' are similar.
6. If Sf5S2, 52'Sj meet in S31', prove that the triangle S1S2S31',
and the two other analogous triangles S2S3SI", 53S1 S2", are similar.
SECTION Y.
SPECIAL APPLICATION OF THE THEORY OF FIGURES.
]DIRECTLY SIMILAR.
10. The Brocard circle.
DEF. 1.-ITf 0 be the circumcentre, and K the syqmmedian point of the triangle ABC, the circle on OK as
diameter is called the -Brocard circle of the triangle.
PEF. 2.-Iffrom 0 _perpendiculars be drawn to the side&
of the triangle ABC, these meet the Brocard circle in three
other points A', B', C', forming' a triangle, which we shall
call -Brocard's First Triangle.