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.

TWO FIGURES DIRECTLY SIMILAR. 17

173

2. If the side BA of the triangle ABC be produced through A
until AB' = BA; and at B' and C be erected perpendiculars to
BA, AC, respectively, meeting in I; AI is perpendicular to the
symnmedian passing through A.
3. If A"/ be the pole of the line BC, with respect to the circuincircie A,,", Ab",) AC'", the feet of the perpendiculars from A",
on the sides of ABC; the area of the triangle A,,", Ab", A,"
1 2A3
(b2 + c2 a2)
4. In the same case prove that the figure A"Ab" A,," AC" is a
parallelogram.
SECTION II.
Two FIGIJRES DIRECTLY SIMILAR.
DEFIN.-_Being given a system of points A, 13, C, D,
If upon the line joining them to a fixed point 0
points A.', 13', C', &c.,...be determined by the condiGA' OF3 OC'-&, h
Gin - A GbB_ =6 C k- tetwo systems of points A, B, C, &c., and A', B', C', &c., are said
to be homothetic, and 0 is called their homothetic centre.
Prop. i.l' The figure homothetic to a right line is
a parallel right line.
20. The fi~gure homzothetic to a circle is a circle.
Dem.-l10 This follows at once from the definition
and _Euc., YI. II.
2'. It is evident from Book VI., Section xx., IProp. 1,
Gor. 4.
Prop. 2.-In two homotheticfigures-10. Two homologous lines are in the constant ratio k. 2'. Two corresponding triangles are similar.
These are evident,