A treatise on the geometry of the circle and some extensions by William J. M'Clelland

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.

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BL OCX triangles PX=OX=r. BPX and Also are equal in every respect (Euc. OXBL is I. 26) a cyclic quadrilateral, therefore Euc. (III. PX=1r\ PL PB is given hence the segment PB is known, and since it ; equal to 0(7, C is determined. In the general case the line A B does not meet the ment is stated : the seg- AB rectangle, having two remaining two on the 3. ] XY of a circle in a given direction such that the area of the quadrilateral A BX Y, where AB is a given diameter, is a maximum. [Draw a diameter YX\ and A YBX' is a rectangle, hence AX' is BY.

The several parts, of a geometrical figure which varies according to a definite law, can always be expressed in terms of the fixed parts of the figure and those quantities which are sufficient to define its position. Take example the figure of Art. 8. In any posi(7, by assuming the triangle to be of given altitude; the variable parts, a, 6, area, and other functions of the sides or angles can be found in terms of the base c vertical angle (7, and altitude. Thus the variables may be regarded as functions of the given parts and the co-ordinates of their position.

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.

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