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.

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A SEQUEL TO EUCLID.

3. The system of circles which are circles of similitude of the
circumcircle and Tucker's circles, respectively, are coaxal.
4. The perpendiculars from K on the sides of the triangle
Ex. 2 are proportional to a2, b2, c2.
3 + b3+ 3 + 3abe
5. The perimeter of Lemoine's hexagon is 2 b2
a2 + b2 + e2
i (a4 + b4 + 4+ a2b2+ b2c2 + c2a2)
and its area - - - ---
(a2 + b2 + c2)2
6. If the cosine circle intersect the sides of ABC in the points
D, D, E', E', F, F', the figures DD'E'F, EE'F'D, FF'D'E are
rectangles; and their areas are proportional to sin 2A, sin 23,
sin 20.
7. In the same case, the diagonal of each rectangle passes
through the symmedian point. This affords a proof of the
theorem, that the middle point of any side, the middle point of
its corresponding perpendicular, and the symmedian point, are
collinear.
8. If the sides of the triangle a/y (fig., Prop. 3) produced,
if necessary, meet the tangents at A, B, C to the circumcircle, six
of the points of intersection are concyclic, and three are collinear.
9. If the distance OK between the circumcentre and symmedian point be divided in the ratio 1: m by the centre of one of
Tucker's circles, and if R, R' be the radii of the circumcircle and
V//2I2,2 + )it222
the cosine circle, the radius of Tucker's circle is
1 + m
10. The square of the diameter of Lemoine's circle is R2 + R'2.
11. If a variable triangle aB3y of given species be inscribed in
a fixed triangle ABC, and if the vertices of aB3y move along the
sides of ABC, the centre of similitude F of a/3y, in any two of its
positions, is a fixed point. (TOWNSEND.)
12. In the same case, if the circumcircle of a3y meet the sides
of ABC in the three additional points a', 3', y'; the triangle a''y'
is given in species, and the centre of similitude F' of it, in any
two of its positions, is a fixed point. (TAYLOR.)
13. Also F, F' are isogonal conjugates with respect to the
triangle.
14. The locus of the centre of the circle a3y is a right line.
15. If through the Brocard points and the centre of any of
Tucker's circles a circle be described, cutting Tucker's circle in
X, Y; prove nX + 'X == nY + 'Y = constant.