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.

230 A SEQUEL TO EUCLID.
tions of.the centre O on the lines PL, PL'; then (a, a'), and the
two other pairs of points similarly determined, are concyclic.
20. Let M, M' be points of the radii OA, OA', such that the
anharmonic ratios (OAML), (OA'M'L') are equal. The projections of O on the lines PM, P'M', and the other pairs of points
determined in the same manner from the chords BB', CC', are
concyclic.
21. The circle of similitude of the three directly similar triangles ABC, FDE, E'F'D' (fig., p. 177) passes through the
Brocard points and Brocard centre of ABC.
22-25. If a, b, c be the points of intersection of the corresponding sides of two equal and directly similar triangles, ABC, A'B'C',
whose centre of similitude is S; then, 1~, if S be the circumcentre of ABC, it is the orthocentre of abe; 2~, if it be the
incentre of ABC, it is the circumcentre of abe; 3~, if S be the
symmedian point of ABC, it is the centroid of abe; 4~, if it be a
Brocard point of ABC, it is a Brocard point of abc.
26. State the corresponding propositions for ABC, and the
triangle formed by the lines joining corresponding vertices of
ABC, A'B'C'.
27. If a cyclic polygon of an even number of sides ABCD,
&c., turn round its circumcentre into the position A'B'C'D', &c.,
each pair of opposite sides of the polygon whose vertices are the
intersection of corresponding sides are parallel.
28. If a variable chord of a circle divide it homographically,
prove that there is a fixed point (Lemoine point) whose distance
from the chord is in a constant ratio to its length.
29. In the same case, prove that there are two Brocard points,
a Brocard angle, a Brocard circle, and systems of invariable
points, and double points.
30. Prove also that the circle can be inverted so that the inverses of the extremities of the homographic chords will be the
extremities of a system of equal chords.