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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Miscellaneous Exercises,
1. If from the symmedian point of any triangle perpendiculars be drawn to its sides, the lines joining their feet are at
right angles to the medians.
2. If ACB be any triangle, CL a perpendicular on AB; prove
that AC and BL are divided proportionally by the antiparallel
to BC through the symmedian point.
3. The middle point of any side of a triangle, and the middle
point of the corresponding perpendicular, are collinear with the
symmedian point.
4. If K be the symmedian point, and G the centroid of the tri.
angle ABC; then-1~, the diameters of the circumcircles of the.
triangles AKB, BKC, CKA, are inversely proportional to the
medians; 2~, the diameters of the circumcircles of the triangles
AGB, BGC, CGA are inversely proportional to AK, BK, CK.
5. If the base BC of a triangle and its Brocard angle be given,
the locus of its vertex is a circle. (NEUBERG.)
Let K be its symmedian point. Through K draw FE parallel
0
Q L
B I LI --

to BO, cutting the perpendicular AL in NM.
2

Make MN equal to