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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. is CY the increment (negative) of BO. Similarly Therefore C'X=CY and the right-angled triangles CC'X CC'Y are equal in AC'C=ACM when A every is in the limit respect, and LAC'C=LBCC'.

Falls outside the triangle ABC become somewhat modified. these angular rela- Take within the angle C. Then from the cyclic quadrilaterals we have 41 for example QRAO and RPBO (Euc. III. 20) LORP = OBPzu& LORQ=OAQ; adding these equations R = OAQ+OBP = C+AOB, AOB = R-C. or Again, since Euc. I. 32, A+ACO = BOC+ABO, by transposing A-BOC=ABO-ACO (1) A BO = RPO since PRBO is cyclic, ACO = QPO since PQCO is cyclic. But and Substituting these values in (1) therefore we have A - BOC= RPO - QPO = P BOC^A - P.

1>Y sin CvPX. ]* 2. The maximum rectangle circle is X and such that F, then NX if BX= CX and inscribed in a given segment of a be drawn at its vertices BC and AC CY= A Y. 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.

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