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文献[1]中给出的命题6如下:已知I是△ABC的内心,r为△ABC的内切圆半径,则有不等式IA~2+IB~2+IC~2≥12r~21这个结论如此简洁、优美,笔者被它深深吸引.经过研究,笔者发现,不等式1可以进一步加强.笔者还对加强不等式进行了一点联想,惊喜地得到一组关于三角形“四心”的不等式.本文约定:△ABC的外接圆半径为R,内切圆半径为r,半周长为p,边BC,CA,AB对应的长度
The proposition 6 given in [1] is as follows: It is known that I is the center of △ ABC and r is the inscribed circle radius of △ ABC, then the inequalities IA ~ 2 + IB ~ 2 + IC ~ 2≥12r ~ 21 The conclusion is so succinct and graceful that the author is deeply attracted to it.After research, I found that Inequality 1 can be further strengthened.The author also made a little suggestion on strengthening inequality, and surprisingly got a group of inequalities about triangle “four hearts ” In this paper, we conclude that the radius of circumcircle of △ ABC is R, the radius of inscribed circle is r, the semi-circumference is p and the length corresponding to BC, CA, AB