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.

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which are collinear), these angles are respectively equal to those
subtended at either Brocard point by the sides of the harmonic
hexagon. (Ex. 11.)
16. If two corresponding points, D, E of two directly similar
figures, F1, F2, be conjugated points with respect to a given circle
(X), the locus of each of the points D, E is a circle.
Dem.-Let S be the double point of F1, F2, and let DE intersect X in L, M. Bisect DE in N. Join SN. Then, from the
property of double points, the triangle SDE is given in species;
therefore the ratio SN: ND is given. Again, because E, D
(hyp.) are harmonic conjugates with respect to L, M, and N is
the middle point of ED, ND2 is equal to the rectangle NM. NL;
that is, equal to the square of the tangent from N to the circle X.
Hence the ratio of SN to the tangent from N to X is given.
Hence the locus of N is a circle, and the triangle SND is given
in species; therefore the locus of D is a circle.
17. If we consider each side of a triangle ABC in succession
as given in magnitude, and also the Brocard angle of the triangle,
the triangle formed by the centres of the three corresponding
Neuberg's circles is in perspective with ABC.
18. If in any triangle ABC triangles similar to its co-symmedian be inscribed, the centre of similitude of the inscribed triangles is the symmedian point of the original triangle.
19. If figures directly similar be described on the sides of a
harmonic hexagon, the middle point of each of its symmedian
lines is a double point for three pairs of figures.
20. If Fi, F2, F3, F4 be figures directly similar described on
the sides of a harmonic quadrilateral, K its symmedian point, K',
K" the extremities of its third diagonal, and if the lines KK',
KK" meet the Brocard circle again in the points H, I; H is the
double point of the figures F1, F3; I of the figures F2, F4.
DEF.-The quadrilateral formed by the four invariable points of
a hormonic quadrilateral is called Brocard'sfirst quadrilateral, and
that formed by the middle points of its diagonals, and the double
points H, I, Brocard's second quadrilateral.
This nomenclature may evidently be extended.