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.

66 A SEQUEL TO EUCLID.
but. the pentagon = 5 A AOB; therefore
twice pentagon = 5Rs;
therefore s (pI + p2 + p3 + p4 + p6) = 5Rs.
Hence P1 +P2 + 3 + _P +P~ = 5R.
Prop. 3.-If a regular polygon of n sides be described
about a circle, the sum of the perpendiculars from the
points of contact on any tangent to the circle egual nR.
Dem.-Let A, B, C, D, E, &c., be the points of contact of the sides of the polygon with the 0, L any
tangent to the 0, and P its point of contact. Now,
the -s from the points A, B, C, &c., on L, are respectively equal to the i-s from P on the tangents at the
same points; but the sum of the -s from P on the tangents at the points A, B, C, &c., = nR (2). Hence the
sum of the is from the points A, B, C, &c., on L = nR.
Cor. 1.-The sum of the is from the angular
points of an inscribed polygon of n sides upon any line
equal n times the -L from the centre on the same line.
Cor. 2.-The centre of mean position of the angular
points of a regular polygon is the centre of its circum.
scribed circle.
For, since there are n points, the sum of the is
from these points on any line equal n times the iL from
their centre of mean position on the line (I., 17);
therefore the.L from the centre of the circumscribed 0
on any line is equal to the 1 from the centre of mean
position on the same line; and, consequently, these
centres must coincide.
Cor. 3.-The sum of the Is from the angular points
of an inscribed polygon on any diameter is zero; or, in
other words, the sum of the _Ls on one side of the diameter is equal to the sum of the Ls on the other side.
Prop. 4. —f a regular polygon of n sides be inscribed
in a circle, whose radius is R, and if P be any point whose
distance from the centre of the circle is R', then the sum
of the squares of all the lines from P to the angular points
of the polygon is equal to n (R2 + RB2).