bisecting lines a line be drawn to the opposite angle of the triangle, it will bisect that angle.

Let ABC be a A; bisect the exterior 4 EAC, ACF in AD, CD, meeting in D, and join DB: the ABC is bisected in BD. Drop the 1rs DE, DF, DG.

B

E

(Depending on Prop. XIV. XXVI. XLVII.)

PROP. T.

CE

D

THEOR.-If from any point within or with

out any rectilinear figure perpendiculars be let fall on every

side, the sum of the squares

of the alternate segments made by them will be equal.

Let ABCD be the figure; E the given pt.; drop the

8 EF, EG, EH, EK; then AF2 + BK2 + DH2 + CG2 = FB2 + KD2 + HC2 + GA2, and so for any number of sides.

(Deducible from Prop. XLVII.)

K

PROP. p. THEOR.-If, in the figure to Prop. XLVII., CF, BK be joined, they will intersect AL, the perpendicular to BC, in the same point.

(See Prop. F.)

PROP. σ. THEOR.-If, in the same figure, perpendiculars be drawn from the points F, K, to BC produced, the line intercepted between B and the perpendicular from F will be equal to the line intercepted between c and the perpendicular from K; and the sum of these lines will be equal to BC.

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