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. 247
141-144. In the same case, if the perpendiculars produced
through the vertices meet the circles again in A", B", C", prove
that1~. AA'B'C' + AA"B"C" = 8 AABC.
2~. Sum of squares of sides of A'B'C', A"B"C"
= 32ABC cot w.
3~. Sum of the cotangents of their Brocard angles
= 2 cot w / (4 - cot2 ). (Ibid.)
145. Prove that the circle in Ex. 60 is coaxal with the ninepoint circle and the Brocard circle.
LONGCHAMPS' CIRCLE, 146-158.-The circle which cuts orthogonally the three circles of Ex. 137 has been studied by M.
LONGCHAMPS, in a paper in the Journal de Mathenatiques Speciales for 1866. The properties which he proves both of a special
nature, and also in connexion with recent geometry, are so interesting that we think it right to give some account of them
here. The demonstrations are in all cases very simple, and
form an excellent exercise for the student. We shall denote
Longchamps' circle by the letter L, and the radical axis of it and
the circumcircle by x. It will be easily seen that the circle is
real only in the case of obtuse-angled triangles.
146. The centre of L is the symmetrique of the orthocentre of
ABC, with respect to its circumcentre.
147. The radius of L is equal to the diameter of the polar
circle of ABC.
148. The circle L is orthogonal to the circles whose centres are
the middle points of the sides of ABC, and whose radii are the
corresponding medians.
149. If I, I' be two isotomic points on any side BC of the
triangle, the circle whose centre is I and radius AI' belongs to a
coaxal system.
150. The line X is the polar of the centroid of the triangle
ABC with respect to L.
151. x is the isotomic conjugate of the Lemoine line of the
triangle ABC, with respect to its sides.
152. x is parallel to the line which joins the isotomic conjugates of the Brocard points of ABC.
153. The trilinear pole of A, with respect to the triangle ABC,