托勒密(ptolemy)定理:圆内接四边形两条对角线之积等于两组对边乘积之和,可推广到任意四边形的情形。凸四边形两条对角线的积的平方,等于两组对边乘积平方的和减去一组对角和的余弦与四边乘积的二倍。已知如图,四边形ABCD的四边为a,b,c,d,对角线为m,n。求证 m~2·n~2=a~2c~2+b~2d~2-2abcdcos(A+C)。证若四边形中至少有一个内角是锐角,例如∠B,设∠ABD=α,∠DBC=β,则n~2=a~2+d~2-2adcosA=b~2+c~ Ptolemy Theorem: The product of the two diagonals of a circle inscribed quadrilateral is equal to the sum of two pairs of side-products, which can be generalized to the case of an arbitrary quadrilateral. The square of the product of the two diagonals of a convex quadrilateral is equal to the sum of the squared product of the two groups, minus the cosine of the group of diagonal sums and the quadrilateral product of the quadrilateral. As shown in the figure, the four sides of the quadrilateral ABCD are a, b, c, d, and the diagonals are m, n. Prove m~2·n~2=a~2c~2+b~2d~2-2abcdcos(A+C). If at least one inner corner of the quadrilateral is an acute angle, such as ∠B, set ∠ABD=α, ∠DBC=β, then n~2=a~2+d~2-2adcosA=b~2+c~