刘保乾——三角形秩序图工程通讯(7) / 四六文摘

刘健著《三正弦不等式》

哈尔滨工业大学出版社，2018

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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id:={[1, 1], [s^2/(16*R*r-5*r^2), 80],

[(ra+rb+rc)/(wb*wc/wa+wc*wa/wb+wa*wb/wc),95], [(4*R^2+4*R*r+3*r^2)/s^2, 79],

[(cos(A)^2+cos(B)^2+cos(C)^2)/(sin((1/2)*A)^2+sin((1/2)*B)^2+sin((1/2)*C)^2),74],

[(cos((1/2)*A)^2+cos((1/2)*B)^2+cos((1/2)*C)^2)/(sin(A)^2+sin(B)^2+sin(C)^2),75],

[(cot(A)+cot(B)+cot(C))/(tan((1/2)*A)+tan((1/2)*B)+tan((1/2)*C)),57],

[(sin((1/2)*A)+sin((1/2)*B)+sin((1/2)*C))/(cos(A)+cos(B)+cos(C)),58],

[(2*(cos(A)^2+cos(B)^2+cos(C)^2))/(cos(A)+cos(B)+cos(C)),54],

[(2*(cos(A)^2+cos(B)^2+cos(C)^2))/(sin((1/2)*A)+sin((1/2)*B)+sin((1/2)*C)),66],

[(2*(sin((1/2)*A)^2+sin((1/2)*B)^2+sin((1/2)*C)^2))/(cos(A)+cos(B)+cos(C)),67],

[(3*(cos(A)^2+cos(B)^2+cos(C)^2))/(cos((1/2)*A)^2+cos((1/2)*B)^2+cos((1/2)*C)^2),76],

[(3*(cos(A)^2+cos(B)^2+cos(C)^2))/(sin(A)^2+sin(B)^2+sin(C)^2),59],

[(3*(cot(A)+cot(B)+cot(C)))/(cot((1/2)*A)+cot((1/2)*B)+cot((1/2)*C)),60],

[(1/3*(tan((1/2)*A)+tan((1/2)*B)+tan((1/2)*C)))*sqrt(3),11],

[(1/4*(sqrt(1-tan((1/2)*B)*tan((1/2)*C))/cos((1/2)*A)+sqrt(1-tan((1/2)*C)*tan((1/2)*A))/cos((1/2)*B)+sqrt(1-tan((1/2)*A)*tan((1/2)*B))/cos((1/2)*C)))*sqrt(2),84],

[(1/6*(a/ma+b/mb+c/mc))*sqrt(3),91], [(1/6*(a/wa+b/wb+c/wc))*sqrt(3), 88],

[(1/6*(sqrt(a/(s-a))+sqrt(b/(s-b))+sqrt(c/(s-c))))*sqrt(2),100],

[(1/9*(wa/(s-a)+wb/(s-b)+wc/(s-c)))*sqrt(3),89],

[(1/9*(cot((1/2)*A)*cos((1/2)*B-(1/2)*C)+cot((1/2)*B)*cos((1/2)*A-(1/2)*C)+cot((1/2)*C)*cos((1/2)*A-(1/2)*B)))*sqrt(3),83],

[(2/9*(ma/a+mb/b+mc/c))*sqrt(3),87],

[(2/9*(cos(A)*cot((1/2)*A)+cos(B)*cot((1/2)*B)+cos(C)*cot((1/2)*C)))*sqrt(3),25],

[(2/9*(sqrt(1-cos(B)*cos(C))+sqrt(1-cos(C)*cos(A))+sqrt(1-cos(A)*cos(B))))*sqrt(3),82],

[1/((cos(A)+cos(B))*(cos(B)+cos(C))*(cos(A)+cos(C))),50],

[(2*(cos(A)^2+cos(B)^2+cos(C)^2))*sqrt(3)/(sin(A)+sin(B)+sin(C)),61],

[(2*(sin((1/2)*A)^2+sin((1/2)*B)^2+sin((1/2)*C)^2))*sqrt(3)/(sin(A)+sin(B)+sin(C)),64],

[(1/9*(cos(A)+cos(B)+cos(C)))*(1/sin(A)+1/sin(B)+1/sin(C))*sqrt(3),85],

[sqrt(3)*(sin((1/2)*A)+sin((1/2)*B)+sin((1/2)*C))/(sin(A)+sin(B)+sin(C)),62],

[9/(4*sin(A)^2+4*sin(B)^2+4*sin(C)^2),52],

[9/(4*sin(A)*sin(B)+4*sin(B)*sin(C)+4*sin(C)*sin(A)),49],

[3/(2*cos(A)+2*cos(B)+2*cos(C)),2],

[3/(2*sin((1/2)*A)+2*sin((1/2)*B)+2*sin((1/2)*C)),3], [(1/16)*R^4/r^4, 12],

[(1/9)*(ma+mb+mc)/r,94],

[(1/2)*((cos(A)+cos(B))^2+(cos(B)+cos(C))^2+(cos(A)+cos(C))^2)/(cos(A)+cos(B)+cos(C)),47],

[(4/3)*(cos((1/2)*A)^4+cos((1/2)*B)^4+cos((1/2)*C)^4)/(cos((1/2)*A)^2+cos((1/2)*B)^2+cos((1/2)*C)^2),24],

[(1/6)*(cos((1/2)*A)*(sin(B)+sin(C))+cos((1/2)*B)*(sin(C)+sin(A))+cos((1/2)*C)*(sin(A)+sin(B)))/(sin((1/2)*B)*sin((1/2)*C)+sin((1/2)*C)*sin((1/2)*A)+sin((1/2)*A)*sin((1/2)*B)),77],

[(1/3)*(cos((1/2)*B)*cos((1/2)*C)+cos((1/2)*C)*cos((1/2)*A)+cos((1/2)*A)*cos((1/2)*B))/(sin((1/2)*B)*sin((1/2)*C)+sin((1/2)*C)*sin((1/2)*A)+sin((1/2)*A)*sin((1/2)*B)),96],

[(3/2)*(cot(A)+cot(B)+cot(C))/(cos((1/2)*A)+cos((1/2)*B)+cos((1/2)*C)),68],

[(3/2)*(cot(A)+cot(B)+cot(C))/(sin(A)+sin(B)+sin(C)),55],

[(1/3)*(sin(A)+sin(B)+sin(C))^2/(sin(B)*sin(C)+sin(C)*sin(A)+sin(A)*sin(B)),56],

[(3/2)*(tan((1/2)*A)+tan((1/2)*B)+tan((1/2)*C))/(sin(A)+sin(B)+sin(C)),69],

[9/((2*(tan((1/2)*A)+tan((1/2)*B)+tan((1/2)*C)))*(sin(A)+sin(B)+sin(C))),37],

[(1/3)*sqrt(3)*(1/(cos((1/2)*A)+cos((1/2)*B))+1/(cos((1/2)*B)+cos((1/2)*C))+1/(cos((1/2)*C)+cos((1/2)*A))),4],

[(1/3)*sqrt(3)*(1/(sin(A)+sin(B))+1/(sin(B)+sin(C))+1/(sin(C)+sin(A))),5],

[(2/3)*sqrt(3)*(1/(2*sin(A)+sin(B)+sin(C))+1/(2*sin(B)+sin(C)+sin(A))+1/(2*sin(C)+sin(A)+sin(B))),53], [(1/6)*sqrt(3)*(1/cos((1/2)*A)+1/cos((1/2)*B)+1/cos((1/2)*C)), 6],

[(1/6)*sqrt(3)*(1/sin(A)+1/sin(B)+1/sin(C)),7], [(2/9)*sqrt(3)*(wa/a+wb/b+wc/c), 86],[(1/6)*sqrt(3)*((cot(B)+cot(C))/(cos(B)+cos(C))+(cot(C)+cot(A))/(cos(A)+cos(C))+(cot(A)+cot(B))/(cos(A)+cos(B))),32],

[(1/36)*sqrt(3)*((cot((1/2)*A)+cot((1/2)*B))^2/cot((1/2)*C)+(cot((1/2)*B)+cot((1/2)*C))^2/cot((1/2)*A)+(cot((1/2)*C)+cot((1/2)*A))^2/cot((1/2)*B)),46],

[(1/18)*sqrt(3)*((cot((1/2)*B)+cot((1/2)*C))/(cos(B)+cos(C))+(cot((1/2)*C)+cot((1/2)*A))/(cos(A)+cos(C))+(cot((1/2)*A)+cot((1/2)*B))/(cos(A)+cos(B))),33],

[(1/6)*sqrt(3)*((tan((1/2)*B)+tan((1/2)*C))/(cos(B)+cos(C))+(tan((1/2)*C)+tan((1/2)*A))/(cos(A)+cos(C))+(tan((1/2)*A)+tan((1/2)*B))/(cos(A)+cos(B))),36],

[(2/3)*sqrt(3)*(cos(A)*tan((1/2)*B)+cos(B)*tan((1/2)*C)+cos(C)*tan((1/2)*A)),28],

[(2/9)*sqrt(3)*(cos((1/2)*A)/(cos(B)+cos(C))+cos((1/2)*B)/(cos(A)+cos(C))+cos((1/2)*C)/(cos(A)+cos(B))),40],

[(2/9)*sqrt(3)*(cos((1/2)*A)/(sin((1/2)*B)+sin((1/2)*C))+cos((1/2)*B)/(sin((1/2)*A)+sin((1/2)*C))+cos((1/2)*C)/(sin((1/2)*A)+sin((1/2)*B))),38],

[(1/3)*sqrt(3)*(cot(A)/(cos(B)+cos(C))+cot(B)/(cos(A)+cos(C))+cot(C)/(cos(A)+cos(B))),21],

[(1/9)*sqrt(3)*(cot((1/2)*A)/(cos(B)+cos(C))+cot((1/2)*B)/(cos(A)+cos(C))+cot((1/2)*C)/(cos(A)+cos(B))),22],

[(1/9)*sqrt(3)*(cot((1/2)*A)/(sin((1/2)*B)+sin((1/2)*C))+cot((1/2)*B)/(sin((1/2)*A)+sin((1/2)*C))+cot((1/2)*C)/(sin((1/2)*A)+sin((1/2)*B))),41],

[(2/9)*sqrt(3)*(sin(A)/(cos(B)+cos(C))+sin(B)/(cos(A)+cos(C))+sin(C)/(cos(A)+cos(B))),23],[(2/3)*sqrt(3)*(sin((1/2)*A)/(sin(B)+sin(C))+sin((1/2)*B)/(sin(C)+sin(A))+sin((1/2)*C)/(sin(A)+sin(B))),42],

[(4/9)*sqrt(3)*(sin((1/2)*A)/(tan((1/2)*B)+tan((1/2)*C))+sin((1/2)*B)/(tan((1/2)*C)+tan((1/2)*A))+sin((1/2)*C)/(tan((1/2)*A)+tan((1/2)*B))),43],

[(2/3)*sqrt(3)*(tan((1/2)*A)^2/(tan((1/2)*A)+tan((1/2)*B))+tan((1/2)*B)^2/(tan((1/2)*B)+tan((1/2)*C))+tan((1/2)*C)^2/(tan((1/2)*C)+tan((1/2)*A))),63],

[(1/9)*sqrt(3)*((cot((1/2)*A)+cot((1/2)*B))*(cot((1/2)*C)+cot((1/2)*A))/(2*cot((1/2)*A)+cot((1/2)*B)+cot((1/2)*C))+(cot((1/2)*A)+cot((1/2)*B))*(cot((1/2)*B)+cot((1/2)*C))/(2*cot((1/2)*B)+cot((1/2)*A)+cot((1/2)*C))+(cot((1/2)*C)+cot((1/2)*A))*(cot((1/2)*B)+cot((1/2)*C))/(2*cot((1/2)*C)+cot((1/2)*A)+cot((1/2)*B))),72],

[(16/27)*sqrt(3)*(cos((1/2)*A)^2*cos((1/2)*B)^2/cot((1/2)*C)+cos((1/2)*B)^2*cos((1/2)*C)^2/cot((1/2)*A)+cos((1/2)*C)^2*cos((1/2)*A)^2/cot((1/2)*B)),48],

[(3/2)*sqrt(3)/(cos((1/2)*A)+cos((1/2)*B)+cos((1/2)*C)),8],

[(1/3)*sqrt(3)*(cot(A)+cot(B)+cot(C)),9], [(3/2)*sqrt(3)/(sin(A)+sin(B)+sin(C)), 10],

[(1/2)*(tan((1/2)*A)+tan((1/2)*B)+tan((1/2)*C))*sqrt(3)/(cos(A)+cos(B)+cos(C)),65],

[(1/3)*sqrt(3)*(cos((1/2)*A)+cos((1/2)*B)+cos((1/2)*C))/(cos(A)+cos(B)+cos(C)),70],

[(1/2)*sqrt(3)*(cot(A)+cot(B)+cot(C))/(sin((1/2)*A)+sin((1/2)*B)+sin((1/2)*C)),73],

[(1/6)*sqrt(3)*(cot((1/2)*A)+cot((1/2)*B)+cot((1/2)*C))/(cos(A)+cos(B)+cos(C)),71],

[(3/8)*sqrt(3)*(cot(B)+cot(C))*(cot(C)+cot(A))*(cot(A)+cot(B)),35],

[(1/72)*sqrt(3)*(cot((1/2)*B)+cot((1/2)*C))*(cot((1/2)*C)+cot((1/2)*A))*(cot((1/2)*A)+cot((1/2)*B)),34],

[3*sqrt(3)/((sin(A)+sin(B))*(sin(B)+sin(C))*(sin(C)+sin(A))), 51],

[(1/9)*sqrt(3)*cot((1/2)*A)*cot((1/2)*B)*cot((1/2)*C),13],

[1/(3*cos(A)+3*cos(B))+1/(3*cos(B)+3*cos(C))+1/(3*cos(C)+3*cos(A)),14],

[1/(3*sin((1/2)*A)+3*sin((1/2)*B))+1/(3*sin((1/2)*B)+3*sin((1/2)*C))+1/(3*sin((1/2)*C)+3*sin((1/2)*A)),15],

[(4/3)*cos(A)^2+(4/3)*cos(B)^2+(4/3)*cos(C)^2, 29],

[(4/3)*sin((1/2)*A)^2+(4/3)*sin((1/2)*B)^2+(4/3)*sin((1/2)*C)^2,30],

[(1/3)*ma/ha+(1/3)*mb/hb+(1/3)*mc/hc,93],

[(2/3)*ma/(hb+hc)+(2/3)*mb/(hc+ha)+(2/3)*mc/(ha+hb),97],

[(1/6)*(rb+rc)/ma+(1/6)*(rc+ra)/mb+(1/6)*(ra+rb)/mc,101],

[(8/3)*cos(A)*sin((1/2)*B)^2+(8/3)*cos(B)*sin((1/2)*C)^2+(8/3)*cos(C)*sin((1/2)*A)^2,31],

[2*cos(A)^2/(sin(B)^2+sin(C)^2)+2*cos(B)^2/(sin(A)^2+sin(C)^2)+2*cos(C)^2/(sin(A)^2+sin(B)^2),26],

[cos(A)^2/cos((1/2)*A)^2+cos(B)^2/cos((1/2)*B)^2+cos(C)^2/cos((1/2)*C)^2,99],

[(1/9)*cot((1/2)*A)*cot((1/2)*B)+(1/9)*cot((1/2)*B)*cot((1/2)*C)+(1/9)*cot((1/2)*C)*cot((1/2)*A),18],

[(2/3)*sin(A)/(sin(B)+sin(C))+(2/3)*sin(B)/(sin(C)+sin(A))+(2/3)*sin(C)/(sin(A)+sin(B)),27],

[(2/3)*tan((1/2)*A)/(tan((1/2)*B)+tan((1/2)*C))+(2/3)*tan((1/2)*B)/(tan((1/2)*C)+tan((1/2)*A))+(2/3)*tan((1/2)*C)/(tan((1/2)*A)+tan((1/2)*B)),45],

[cos(A)^2/(sin(B)*sin(C))+cos(B)^2/(sin(C)*sin(A))+cos(C)^2/(sin(A)*sin(B)),98],

[(4/9)*sin(A)*sin(B)/(cos(A)+cos(B))+(4/9)*sin(B)*sin(C)/(cos(B)+cos(C))+(4/9)*sin(C)*sin(A)/(cos(A)+cos(C)),44],

[(4/3)*sin((1/2)*A)*sin((1/2)*B)/(cos(A)+cos(B))+(4/3)*sin((1/2)*B)*sin((1/2)*C)/(cos(B)+cos(C))+(4/3)*sin((1/2)*C)*sin((1/2)*A)/(cos(A)+cos(C)),39],

[tan((1/2)*A)^2*ha^2/wa^2+tan((1/2)*B)^2*hb^2/wb^2+tan((1/2)*C)^2*hc^2/wc^2,78],

[(4/9)*mb*mc/(b*c)+(4/9)*mc*ma/(a*c)+(4/9)*ma*mb/(a*b),90],

[(1/9)*wb*wc/((s-b)*(s-c))+(1/9)*wc*wa/((s-c)*(s-a))+(1/9)*wa*wb/((s-a)*(s-b)),92],

[1/(4*cos((1/2)*A)*cos((1/2)*B))+1/(4*cos((1/2)*B)*cos((1/2)*C))+1/(4*cos((1/2)*C)*cos((1/2)*A)),17], [1/(4*sin(A)*sin(B))+1/(4*sin(B)*sin(C))+1/(4*sin(C)*sin(A)), 19],

[1/(12*sin((1/2)*A)*sin((1/2)*B))+1/(12*sin((1/2)*B)*sin((1/2)*C))+1/(12*sin((1/2)*C)*sin((1/2)*A)),20], [1/(6*sin((1/2)*A))+1/(6*sin((1/2)*B))+1/(6*sin((1/2)*C)), 16],

[(2/3)*sqrt(1-sin(B)*sin(C))+(2/3)*sqrt(1-sin(C)*sin(A))+(2/3)*sqrt(1-sin(A)*sin(B)),81]};

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