By William J. M'Clelland
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Extra info for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.
Circle Euc. III. 15). EXAMPLES. 1. Having given the base and when the area is a maximum or locus of vertex of a triangle find ; minimum. [Let the locus be a curve of any order and it is readily seen I. ] (Euc. 2. In Ex. 1 when is the sum of the sides a minimum or maximum ? [Let locus C and MN. C' be two points the indefinitely near to each other on and C' Y perpendiculars to AC' and EC CX Draw respectively. Then since in the triangle finitely small, ACX nearly equal to AX. is A CX, X is a right angle and A approximately a right angle and Hence inde- AC C'X=AC'-AX=AC'-AC.
As the altitude thus receives indefinitely small increments so does the area. Let AB1&& the base of a triangle and any curve GG^G^ the locus of its vertex. * See Burnside and Panton's Theory of Equations, Art, c 7. MAXIMUM AND MINIMUM. 34 In the figure as the vertex approaches C on the curve from left to right the intercept made by the per- AX pendicular may position, since if also known. Thus while be taken as AX AX is the known co-ordinate of its the position of is continues to receive positive increit are ments, the area, altitude, and other functions of sometimes decreasing, as from increasing, as from Cl to G2 G to Ov and sometimes .
Be divided into is a any maximum when the line is bisected. For Euc. hence ab is is a (^^ = ("-)* = const ab + (II. 5) maximum when a b = or when a = " b. COR. 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.
A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples. by William J. M'Clelland