AF produced will bisect BC in G, and, if AB = n. AD, find the ratio of AF to AG.

91. If any line be drawn cutting the sides BC, AC, AB of a triangle in a, b, c, shew that Ab.Bc. Ca- Ac. Ba.Cb; and, conversely, if this equality obtain, the points a, b, c lie in one line.

92. If one side of a triangle, AB, be produced to D, and the other, AC, shortened by CF, equal to BD, the line DF will divide BC in the ratio of AC to AB, and BC will divide DF in the ratio of AF to AD.

93. Lines are drawn from the angles of a triangle ABC, through any point, to meet the sides in a, b, c, and through these points a circle is described, cutting the same sides again in a', b', c': shew that Aa', Bb', Cc meet in a point.

94. If through a given point within a triangle lines be drawn from the angles to the opposite sides, and the points of section be joined, the three first drawn lines will be harmonically divided: and, conversely, if any line AH be divided harmonically in G, D, and F be a point in any other line AB, the intersections E, C, of FG, FD, with BD, BH, will be in a line.

95. If X be an object inaccessible to an observer A, shew that, taking in the direction XA any point B, and in any other direction through B any two points C, D, and È the point of intersection of the lines AD, CX, AB. CD. EA BC.DE-AE. CD'

AX =

96. Under the same circumstances as in [95] describe any triangle ABC, and through X draw any line, cutting AČ, BC in D, E: then, if the line through C and the AB. AF

intersection of AE, BD, cut AB in F, AX : = BF-AF

97. The sides AB, AC, BC, of an inscribed triangle are produced to cut a given line in m, n, p; if tm, tn, tp, be the lengths of the tangents drawn from m, n, p, to the circle, shew that tm.tn.tp=Am. Bp. Cn = An. Bm.Cp.

98. In every inscribed triangle, the intersection of the sides and tangents at the angles opposite are in a line.

99. In every inscribed hexagon, the intersections of the opposite sides taken two and two are in one line.

100. If a hexagon be described about a circle, the lines joining the opposite angles intersect in a point.

By the same Author.

THE ELEMENTS OF ALGEBRA.

Fourth Edition. Price 4s. 6d.

ARITHMETIC.

Third Edition. Price 4s. 6d.

These are now in use as School Books at Eton, Harrow, Winchester, Rugby, Shrewsbury, and most other public Schools.