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.

68

A SEQUEL TO EUCLID.

Dem.-From the last Proposition, we have
BA' AOB
AOC'
CB' BOO
B'A BOA'
AC' AOO
(CB BOC'
Hence, multiplying out, we get the product equal to
unity.
Cor. This may be written
AB'. BC'. CA' = A'B. B'. OA.
The symmetry of this expression is apparent. Expressed in words, it gives the product of three alternate segments of the sides equal to the product of the
three remaining segments.
Prop. 3.-If two parallel lines be intersected by three
concurrent transversals, the segments intercepted by the
transversals on the parallels are c
proportional.
Let the Its be AB, A'B', and the
transversals CA, CD, OB; then A D B
AD: DB:: A'D': D'B'.
D\
Dem.- The triangles ADO, A B
A'D'C are equiangular;
therefore AD: DC: A'D: D'O.
In like manner, DC: DB D'C: D'B';
therefore ex aeuadi AD: DB:: A'D': DB'.
Cor.-If from the points D, D' we draw two ILs
DE, D'E' to AC, and two Is DF, D'' to BC; then

DE: DF:: "D'E': D''.