By William J. M'Clelland
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Additional resources for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.
The maximum triangle of given perimeter is equilateral. ] * ; ; MAXIMUM AND MINIMUM. 30 The maximum parallelogram 7. is of given perimeter and angles equilateral. 2 p z denote the perpendiculars from any point If 8. is on the , sides of a triangle, the maximum value of Pip^ps then the centroid of the triangle. (By Ex. ) Since 4^abp p 2 ^(ap l + bp 2 ) 2 - (ap l bp 2 ) 2 for any [Otherwise thus 011 the base c, p^p^ is maximum when ap l point bp 2 vanishes, since ap l + bp 2 equals 2A. Then is the middle point of the base.
Tangents maximum rectangle that can be inscribed in the therefore greater than any other X'N. Hence is from the symmetry of the figure the rectangle on the side [For triangle is the BCN, and XY greater than that on X'Y', and a fortiori greater than that on X"Y". * Hence, the maximum the area of the triangle. parallelogram inscribed in a triangle is half MAXIMUM AND MINIMUM. 28 Let The construction of the maximum rectangle is as follows be drawn perpendicular to OL, the diameter of the circle Join OX and let it meet BL in P.
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. ] circle. 6. Having given two opposite sides A B and CD of a quadrilateral and the diagonals CA and BD, to construct it so that the area may be a maximum. AB be fixed and draw C' and D consecutive positions of C and /).
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