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.

THEORY OF ISOGONAL AND ISOTOMTIC POINTS. 171
CO..- a y =z = 2
CoR.I.- -a b - c - a+b62 Ac6
Coin. 2.- The symmedians of a triangle are the medians
of its anti~parallels.
Con. 3.-The symmiedian AS divides BC in the. ratio
AB2: AC'.
CoR,. 4.-The multi~ples for which K is the mean centre
of the _points A, B, C are a", b2, c2.
CoR. 5.-The _point K is the isogonal conjugate of the
ceitroid of the triangle.
Prop. 6.-The symmedians pass through the poles of
the sides of the triangle ABC with respect to its circumcircle.
Dem.-iLet A' be the pole of BC. Let fall perpendiculars ATF, A'G; then A A'F KG: sin A'BF
sinuA'CG,or::sin ACB: sin ABC;.-. ATF: A'G:: AB:AC: perpendicular from K on AB:perpendicular
from K on AC. Hence the points A, K, A' are collinear.
Coin. 1.-The polar of K is the axis of perspective of
the triangle ABC, and its reci[procal, with respect to the
eircumocircle.
CoR 2.-The tangent at A to the circumcircle, and the
symmedian AS, are harmonico conjugates with res~pect to the
sides BA, AC of the triangle.
Prop. 7.-The sum of the squares of the' distances of
K from the sides of ABC is. a minimum.
Dem.-Let x, y, z be the distances of any point
whatever from the sides of ABC, and let A denote
its area. We have the identity
(x 2 + p2 + z2 )(a2 +i b2 + c2 ) - (ax + by + cz)2
= (ay - bx )+(bz -cy)2~+(cx -az)2;
butax + by + cs = 2~A. Consequently 2'+ p2 + z2 has its