刘保乾——三角形秩序图工程通讯(8)

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刘健著《三正弦不等式》

哈尔滨工业大学出版社，2018

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本书介绍了作者在几何不等式领城的一项发现——三正弦不等式，着重讨论了它的应用，由此推导出了大量涉及三角形的不等式，其中包含许多著名结果，如Wolstenholme不等式.Kooi不等式、Klamkin惯性极矩不等式、Erdös-Mordell不等式、Neuberg-Pedoe不等式、Cerretsen不等式、林鹤一不等式、锐角三角形的Walker 不等式、推广的Euler不等式.作者在本书中还针对相关结果提出了大量经过计算机验证的不等式猜想，可供有兴趣的读者研究.

附录A介绍了有关三角形与一点的几何变换理论(便于书中一些几何不等式的推导)，附录B介绍了作者建立的一些涉及多个三角形不等式的结果，其中包含了三正弦不等式的推广.

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1．坐标数据

{[1,[1], [28, 57, 79, 25, 46]], [2, [28, 58, 27], [63, 3, 35, 62, 22]], [3, [57, 2,4, 40, 80], [29, 5, 32, 36]], [4, [28, 58], [63, 3, 35, 62, 15, 22]], [5, [56,63, 3, 35, 22], [38, 12]], [6, [79, 46], [58, 47, 11, 20, 21, 48, 23]], [7,[60, 38, 62, 13, 14, 18, 12], [69, 50]], [8, [63, 75, 22], [38]], [9, [65, 52],[9]], [10, [79], [58, 11, 78, 16]], [11, [28, 57, 6, 10, 24], [29, 33, 26, 15,17]], [12, [54, 5], [7, 43]], [13, [47, 40, 80, 20], [55, 37, 7]], [14, [47,40, 67, 48], [55, 37, 7]], [15, [4, 11, 20], [69, 45, 50]], [16, [25, 10], [60,47, 67, 23]], [17, [35, 40, 11, 22], [54, 45]], [18, [40, 80, 27], [37, 7, 43,77]], [19, [40, 80, 27], [44]], [20, [6, 78, 24], [13, 15, 27]], [21, [6, 81,24], [63, 35, 40, 51, 22]], [22, [57, 2, 4, 80, 21, 23], [29, 5, 32, 8, 26,17]], [23, [6, 81, 16, 24], [63, 40, 64, 51, 22]], [24, [79, 46], [58, 11, 20,21, 48, 23]], [25, [1], [81, 16]], [26, [35, 40, 11, 22], [54, 45]], [27, [81,20, 48], [56, 2, 34, 51, 18, 19, 49]], [28, [1], [56, 2, 4, 11]], [29, [56, 63,3, 35, 11, 22], [54]], [30, [70], [42, 44]], [31, [34, 43, 69, 50], [73, 39,70, 74, 71, 52]], [32, [3, 35, 22], [54, 37, 77]], [33, [35, 11], [73, 39, 41,70, 74, 52]], [34, [76, 40, 80, 27], [59, 31, 68]], [35, [57, 2, 4, 21], [29,5, 32, 33, 26, 75, 17]], [36, [56, 3, 47], [43, 75]], [37, [60, 32, 38, 62, 13,14, 18], [69, 50]], [38, [5, 8, 49], [55, 37, 7]], [39, [31, 33, 64, 77],[65]], [40, [21, 23], [56, 3, 34, 26, 13, 14, 17, 18, 19, 49]], [41, [33, 66],[44]], [42, [30, 73, 59, 71], [65]], [43, [36, 12, 18, 45], [31]], [44, [30,41, 19], [65]], [45, [26, 15, 17], [43, 66]], [46, [1], [76, 6, 67, 78, 24]],[47, [6, 81, 16], [36, 64, 51, 13, 14, 49]], [48, [6, 78, 24], [80, 14, 27]],[49, [47, 40, 80, 27], [38]], [50, [37, 7, 15], [59, 31, 68]], [51, [47, 80,21, 23, 27], [62, 75]], [52, [53, 31, 33, 64, 66, 77], [9]], [53, [57], [52]],[54, [29, 32, 26, 17], [12]], [55, [60, 38, 62, 13, 14], [69, 72, 77]], [56,[28, 58, 40, 80, 27], [29, 5, 36, 62]], [57, [1], [63, 53, 3, 35, 62, 11, 22]],[58, [76, 6, 10, 24], [56, 2, 4]], [59, [61, 34, 64, 69, 72, 50, 77], [42]],[60, [16], [55, 61, 37, 7]], [61, [60, 81, 67], [59, 68]], [62, [56, 57, 2, 4,67, 51], [55, 37, 7]], [63, [57, 2, 4, 80, 21, 23], [29, 5, 8]], [64, [47, 67,23], [59, 39, 68, 70, 74, 52]], [65, [39, 42, 74, 44], [9]], [66, [45], [41,70, 74, 52]], [67, [46, 16], [61, 62, 64, 75, 14]], [68, [61, 34, 64, 69, 72,50, 77], [73, 71]], [69, [55, 37, 7, 15], [59, 31, 68]], [70, [31, 33, 64, 66,77], [30]], [71, [31, 68], [42]], [72, [55], [59, 68]], [73, [31, 33, 68],[42]], [74, [31, 33, 64, 66, 77], [65]], [75, [35, 36, 67, 51], [8]], [76,[46], [58, 34]], [77, [55, 32, 18], [59, 39, 68, 70, 74, 52]], [78, [46, 10],[81, 20, 48]], [79, [1], [6, 10, 24]], [80, [81, 48], [56, 63, 3, 34, 51, 13, 18,19, 22, 49]], [81, [25, 78], [61, 47, 80, 21, 23, 27]]}

2．几何量及ID号

{[1,1], [(ha+hb+hc)/(ma+mb+mc), 56], [(ha+hb+hc)/(ra+rb+rc), 29],[(ha+hb+hc)/(wa+wb+wc), 28], [(ma+mb+mc)/(ra+rb+rc), 57],[(wa+wb+wc)/(ra+rb+rc), 63], [(ma^2*cos(B-C)+mb^2*cos(A-C)+mc^2*cos(A-B))/s^2,53], [(1/3*((s-a)/wa+(s-b)/wb+(s-c)/wc))*sqrt(3), 46],[(1/6*(sqrt(1-tan((1/2)*B)*tan((1/2)*C))+sqrt(1-tan((1/2)*C)*tan((1/2)*A))+sqrt(1-tan((1/2)*A)*tan((1/2)*B))))*sqrt(6),25], [(4/9*(wa/(b+c)+wb/(c+a)+wc/(a+b)))*sqrt(3), 47], [sqrt(3)/(cot(A)+cot(B)+cot(C)),55], [(cos(A)+cos(B))*(cos(B)+cos(C))*(cos(A)+cos(C)), 30],[(8/27*(cos((1/2)*A)^2+cos((1/2)*B)^2))*(cos((1/2)*B)^2+cos((1/2)*C)^2)*(cos((1/2)*C)^2+cos((1/2)*A)^2),40], [(8/27*(sin(A)^2+sin(B)^2))*(sin(B)^2+sin(C)^2)*(sin(A)^2+sin(C)^2), 41],[cos((1/2)*B-(1/2)*C)*cos((1/2)*A-(1/2)*C)*cos((1/2)*A-(1/2)*B), 54],[sqrt(3)*s/(ra+rb+rc), 58],[sqrt(3)*s/(ha*wb*wc/(hb*hc)+hb*wc*wa/(hc*ha)+hc*wa*wb/(ha*hb)), 76],[(wa-r)*(wb-r)*(wc-r)/R^3, 73], [(8*(wa-2*r))*(wb-2*r)*(wc-2*r)/R^3, 74], [ha*hb*hc/(ma*mb*mc),59], [ra*rb*rc/(ma*mb*mc), 60], [wa*wb*wc/(ma*mb*mc), 61], [2*r/R, 7],[9*r/(ma+mb+mc), 62], [(1/9)*sqrt(3)*(a/R+b/R+c/R), 2],[(1/3)*sqrt(3)*(ha/s+hb/s+hc/s), 3],[(1/2)*sqrt(3)*((ha-r)/s+(hb-r)/s+(hc-r)/s), 4], [(5/9)*sqrt(3)*(-(-s+a)/(4*R-ra)-(-s+b)/(4*R-rb)-(-s+c)/(4*R-rc)),5],[(8/27)*sqrt(3)*(cos((1/2)*A)^2*sin(A)+cos((1/2)*B)^2*sin(B)+cos((1/2)*C)^2*sin(C)),32], [(4/9)*sqrt(3)*(sin(A)*cos(A)+sin(B)*cos(B)+sin(C)*cos(C)), 31],[(16/81)*sqrt(3)*(sin(A)^4/tan((1/2)*A)+sin(B)^4/tan((1/2)*B)+sin(C)^4/tan((1/2)*C)),33],[(16/27)*sqrt(3)*(cos(A)*cos((1/2)*A)^2*sin(A)+cos(B)*cos((1/2)*B)^2*sin(B)+cos(C)*cos((1/2)*C)^2*sin(C)),34],[(32/81)*sqrt(3)*(sin(A)*cos((1/2)*B)^2*cos((1/2)*C)^2+sin(B)*cos((1/2)*C)^2*cos((1/2)*A)^2+sin(C)*cos((1/2)*A)^2*cos((1/2)*B)^2),35],[(4/9)*sqrt(3)*(sin(A)*sin(B)/(sin(A)+sin(B))+sin(B)*sin(C)/(sin(B)+sin(C))+sin(A)*sin(C)/(sin(C)+sin(A))),36],[(4/9)*sqrt(3)*(sin(A)*sin(B)/cot((1/2)*C)+sin(B)*sin(C)/cot((1/2)*A)+sin(A)*sin(C)/cot((1/2)*B)),37], [(2/3)*sqrt(3)*(sin(A)*tan((1/2)*B)/cot((1/2)*C)+sin(B)*tan((1/2)*C)/cot((1/2)*A)+sin(C)*tan((1/2)*A)/cot((1/2)*B)),38],[(16/27)*sqrt(3)*(sin(A)^2*sin(B)^2/cot((1/2)*C)+sin(B)^2*sin(C)^2/cot((1/2)*A)+sin(A)^2*sin(C)^2/cot((1/2)*B)),39], [(2/9)*sqrt(3)*(cos((1/2)*A)+cos((1/2)*B)+cos((1/2)*C)), 6],[(1/3)*sqrt(cos(A)+cos(B)+cos(C))*sqrt(6), 81], [8*cos(A)*cos(B)*cos(C), 9],[(64/27)*cos((1/2)*A)^2*cos((1/2)*B)^2*cos((1/2)*C)^2, 26],[64*sin((1/2)*A)^2*sin((1/2)*B)^2*sin((1/2)*C)^2, 42], [3*sqrt(3)*r/s, 8], [3*a*b*c/(a^3+b^3+c^3),64], [(8/27)*ha*hb*hc/R^3, 65], [(8/27)*ma*mb*mc/R^3, 66], [27*r^3/(ha*hb*hc),67], [27*r^3/(ra*rb*rc), 68], [27*r^3/(wa*wb*wc), 69], [(8/27)*ra*rb*rc/R^3,70], [(1/9)*sqrt(3)*(sin(A)+sin(B))*(sin(B)+sin(C))*(sin(C)+sin(A)), 43],[24*sqrt(3)*r^3/(a*b*c), 71], [3*sqrt(3)*wa*wb*wc/s^3, 72],[(8/27)*sin(A)*(sin(B)+sin(C))*sin(B)*(sin(C)+sin(A))*sin(C)*(sin(A)+sin(B)),44], [(8/9)*sqrt(3)*wa*wb*wc/(a*b*c), 75],[(2/3)*sqrt(cos((1/2)*A)^2+cos((1/2)*B)^2+cos((1/2)*C)^2), 79],[(4/9)*cos((1/2)*A)^2+(4/9)*cos((1/2)*B)^2+(4/9)*cos((1/2)*C)^2, 10],[(4/9)*sin(A)^2+(4/9)*sin(B)^2+(4/9)*sin(C)^2, 11],[(1/3)*ga/wa+(1/3)*gb/wb+(1/3)*gc/wc, 78],[(1/3)*ha/ma+(1/3)*hb/mb+(1/3)*hc/mc, 51],[(5/9)*ha/(4*R-ra)+(5/9)*hb/(4*R-rb)+(5/9)*hc/(4*R-rc), 13], [(5/3)*r/(4*R-ra)+(5/3)*r/(4*R-rb)+(5/3)*r/(4*R-rc),14], [(5/3)*(ha-2*r)/(4*R-ra)+(5/3)*(hb-2*r)/(4*R-rb)+(5/3)*(hc-2*r)/(4*R-rc),15], [(2/3)*(ha-2*r)/(ha-r)+(2/3)*(hb-2*r)/(hb-r)+(2/3)*(hc-2*r)/(hc-r), 16],[(1/3)*(ha-r)/R+(1/3)*(hb-r)/R+(1/3)*(hc-r)/R, 17], [(5/6)*(ha-r)/(4*R-ra)+(5/6)*(hb-r)/(4*R-rb)+(5/6)*(hc-r)/(4*R-rc),18], [-(1/6)*(-ra+r)/(2*R-ha)-(1/6)*(-rb+r)/(2*R-hb)-(1/6)*(-rc+r)/(2*R-hc),19], [(4/3)*cos(B)*cos(C)+(4/3)*cos(A)*cos(C)+(4/3)*cos(A)*cos(B), 52],[(4/9)*cos((1/2)*A)*cos((1/2)*B)+(4/9)*cos((1/2)*B)*cos((1/2)*C)+(4/9)*cos((1/2)*C)*cos((1/2)*A),20],[(16/27)*cos((1/2)*A)^2*cos((1/2)*B)^2+(16/27)*cos((1/2)*B)^2*cos((1/2)*C)^2+(16/27)*cos((1/2)*C)^2*cos((1/2)*A)^2,21], [(16/27)*cos((1/2)*A)^2*sin(A)^2+(16/27)*cos((1/2)*B)^2*sin(B)^2+(16/27)*cos((1/2)*C)^2*sin(C)^2,45], [(4/9)*sin(B)*sin(C)+(4/9)*sin(A)*sin(C)+(4/9)*sin(A)*sin(B), 12],[(4/3)*sin((1/2)*A)*sin((1/2)*B)+(4/3)*sin((1/2)*B)*sin((1/2)*C)+(4/3)*sin((1/2)*C)*sin((1/2)*A),22], [(1/3)*sin(A)*sin(B)/cos((1/2)*C)^2+(1/3)*sin(B)*sin(C)/cos((1/2)*A)^2+(1/3)*sin(A)*sin(C)/cos((1/2)*B)^2,49],[(16/9)*sin(A)*sin(B)*sin((1/2)*C)^2+(16/9)*sin(B)*sin(C)*sin((1/2)*A)^2+(16/9)*sin(A)*sin(C)*sin((1/2)*B)^2,50], [(1/3)*hb*hc/(mb*mc)+(1/3)*hc*ha/(mc*ma)+(1/3)*ha*hb/(ma*mb), 77],[(s-b)*(s-c)/(wb*wc)+(s-c)*(s-a)/(wc*wa)+(s-a)*(s-b)/(wa*wb), 48],[(2/3)*cos(A)+(2/3)*cos(B)+(2/3)*cos(C), 23],[(1/3)*cos((1/2)*B-(1/2)*C)+(1/3)*cos((1/2)*A-(1/2)*C)+(1/3)*cos((1/2)*A-(1/2)*B),27], [(2/3)*sin((1/2)*A)+(2/3)*sin((1/2)*B)+(2/3)*sin((1/2)*C), 24], [(1/3)*sqrt(cos(B)+cos(C))+(1/3)*sqrt(cos(A)+cos(C))+(1/3)*sqrt(cos(A)+cos(B)),80]}

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