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.

MISCELLANEOUS EXERCISES. 239
persetive are collinear; and the line of collinearity bisects FG at
right angles.
4~. The lines AG, CG, BF, DF are tangential to a circle concentric with the circumcircle.
5~. If X, Y, Z, W be the feet of perpendiculars from E on the
sides of ABCD, E is the mean centre of X, Y, Z, W.
6~. The sides of the quadrilateral XYZW are tangential to a
circle concentric with the circumcircle.
52-55. If R, R' be the symmedan points of the triangles ABC,
ADC; S, S' of the triangles BCD, DAB; then1~. The quadrilaterals ABCD, S'RSR', have a common harmonic triangle.
2~. The four lines RS, S'R', AD, BC, are concurrent.
3~. If E' be the pole of AC; E" of BD, the three pairs of
points A, C; S', S; E, E", form an involution of which E, E"
are the double points.
4~. If through E a parallel to its polar be drawn, meeting the
four concurrent lines AD, S'R', RS, BC in the points A, A, v, p,
the four intercepts Au, rE, EY, vp are equal; and a similar
property holds for the intercepts on the parallel made by the
lines AB, S'R, R'S, CD.
56. If two triangles, formed by two triads of corresponding
points of three figures, F1, F2, F3, directly similar, be in perspective, the locus of their centre of perspective is the circle of
similitude of F1, F2, F3. (TARRY.)
57. If the symmedian lines AK, BK, CK, &c., of a harmonic
polygon of an odd number of sides, be produced to meet the circumcircle again in the points A', B', C', &c., these points form
the vertices of another harmonic polygon-; and these two polygons are co-symmedian, and have the same Brocard angles, Brocard points, Lemoine circles, cosine circles, &c.
58. If three similar isosceles triangles BEA', CAB', ABC' be
described on the sides of a triangle ABC, prove that the axis of
perspective of the triangles ABC, A'B'C' is perpendicular to the
line joining their centre of perspective to the circumcentre of
ABC. (M'CAY.)
59. In the same, if perpendiculars be let fall from A, B, C on
B'C', C'A', A'B', prove that their point of concurrence is col