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By William J. M'Clelland

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Extra resources for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.

Example text

The polygon 2. one is Ex. ) of least perimeter that can be inscribed in a given that whose angles are bisected externally by its sides. (By The base and area 3. least 4. of a triangle being given, the perimeter is the triangle is isosceles. ] If from 0, the point of intersection of the diagonals of a cyclic when quadrilateral, perpendiculars are 1\ Q R S joined, y t drawn the quadrilateral to the sides PQRS is of and their feet minimum perimeter. 4a. If points P, ', R', S' be taken on the sides of the given quadrilateral, such that P'Q', Q'R'y R'S' are parallel to PQ, QR, RS, then P'ti' is parallel to PS and the perimeters of the quadrilaterals [Euc.

The diagonals of a quadrilateral are 9 and 10 feet and two opposite sides 5 and 3 feet find when its area is a maximum. 6. ; 10. and Theorem. Having given the locus of the vertex a the base line L A B of a triangle meeting the base pro- AC+BO is a minimum duced, the sum of the sides L is the external bisector of the vertical angle. Let fall AE' and when a perpendicular BL and make B'L = BL. Join G be its intersection with L. Take any let other point P on the line and join AP and B'P. MAXIMUM AND MINIMUM.

Of intersection is P of two circles draw a line a minimum. [This reduces to describe a circle touching the two given ones at are in a line. ] Theorem. // a right line two parts a and b, their rectangle 14-. 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.

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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


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