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.
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.
34' = For the side 3, (2) 34' we = (2) 4, and the order (3) get -31' 2T+T2' 34 = 0, . ] . 1 opposite contacts with a CHAPTER II. MAXIMUM AND MINIMUM INTRODUCTION. When the base and vertical angle of a triangle are of the vertex is a segment of a circle locus the given on the described base, containing an angle equal to the Let a number of trivertical angle. (Euc. III. ) 8. angles be constructed satisfying the given conditions, it will be observed that as the vertex recedes from and either extremity of the base the altitude and area both increase up to a certain point, after which they begin to diminish.
3. Given the vertex A of a triangle fixed, the angle A in magnitude and the base angles moving on fixed lines intersecting in ; ABC of minimum area. to construct the triangle [By taking two consecutive positions as in figure, we have AB. AC=AB' AC' and LBAB'=LCAC'. Hence AB :AB' = AC' and the triangles BAB' and CA C' : AC, are similar (Euc. VI. 6). ] LABO = LAC'0 = ACO in the limit In the required position the sides A B and AC are equally inclined to the given lines. Here again we have an illustration of the Therefore symmetry of the figure when the triangle is minimum.