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In the particular case when the curve whose centre is at the triangle is isosceles. is a This MAXIMUM AND MINIMUM. 24 may be stated otherwise. When the vertical and altitude of a triangle are given, the base and angle area are both minima when the triangle is isosceles. property On account of its importance an independent proof of this property of the isosceles triangle is given. Let ABO be an isosceles triangle and A'B'C any other, having the same vertical angle and altitude CM. Now BC>B'C(Euc.

The continued product of the segments of a line a maximum when the parts are equal. EXAMPLES. 1. Through any point P on the base of a triangle parallels PX and P-Tare drawn PXCYi* a to the opposite sides maximum when ; the area of the parallelogram is bisected at P. AB BP Y the base are constant in species, [For the triangles A I'X and But the area of the parallelogram hence PX. PYccAP. BP. = PX. 1>Y sin CvPX. ]* 2. The maximum rectangle circle is X and such that F, then NX if BX= CX and inscribed in a given segment of a be drawn at its vertices BC and AC CY= A Y.

Take two consecutive positions of the secant that ABCD^AB'C'D, and Now since join AB, AB\ DC, DC', and ABCD = AB'C'D it follows that BB'CC'=ABB' + DCC' or BB C+ CB'C = BB'A + CC'D. f BC &nd OB' C 1 B'C. f such MAXIMUM AND MINIMUM. 38 Transposing we have BB'C- BEA = CC'D - CB'C', or since twice the area of a triangle is the product of two sides x the sine of the included angle ; in the limit this relation becomes BB'(BC* diameter diameter ' Hence if AB=a, but from similar triangles BB'/CC'=OBIOC.

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