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.

236

A SEQUEL TO EUCLID.

34, A', C' are corresponding points of figures F2, F1 directly
similar described on the sides BC, AB of a given triangle: if AA',
CC' be parallel, the loci of the points A, C are circles.
Dem.-Let S be the double point of the figures, and D the
point of F2, which corresponds to C in F1. Join DA', and produce CC' to meet it in E. Now, since S is the double point of
the figures ABCC', BCDA', the triangles SCC', SDA' are equiangular; therefore the angle SCC' is equal to SDA'. Hence the
points S, D, E, C, are concyclic; therefore the angle DEC is
equal to the supplement of DSC; that is, equal to ABC; therefore DA'A is equal to ABC, and is given. Hence the locus of
A' is a circle.
35. The Brocard angles of the triangles ABC, BCA' are equal.
Dem.-Let the circle about the triangle DA'A cut AB in F.

C'

Join DF, CF; then the angle DFB is equal to DA'A [Euc. III.,
xxIL]; therefore it is equal to BCD. Hence the triangles BCA,
BCF, BCD are equiangular. Hence the circle which is the locus
of the vertex when the base BC is given, and the Brocard angle