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三角形中的一个点,居然导致经济学家致电几何学家,这里面藏有多少不为人知的奥秘?看似平凡的三角形却让众多数学家得出很多不平凡的结论,这里面真的暗藏玄机?一个小小的图形,却是数学家的万花筒,只要稍微一动,就会绽放光彩,如果信手拆开,原来只不过是几片涂有颜色的纸片而已……一天,几何学家佩多教授接到了某位经济学家打来的电话.这位经济学家向他请教:如果正三角形内有一个点P,那么,不管P的位置在三角形内如何变动,P到三角形三边距离之和是否总是不变的呢?佩多教授马上给了让他满意的答复.如图1,把△ABC分成△PAB,△PBC,△PCA.A F B D C E Pz x y图1用x,y,z分别记P到△ABC三边的距离.由于△ABC三条边相等,设为a,则S△PBC=12ax,S△PAC=12ay,S△PAB=12az,于是x+y+z=2S△ABC a①.
One point in the triangle, actually led economists to call the geometry, which hidden inside the mystery of how many unknown? Seemingly ordinary triangle allows many mathematicians come to many extraordinary conclusions, which really hidden inside Mystery? A small figure, but it is mathematician kaleidoscope, as long as a move, it will bloom, if the letter open, the original is only a few pieces of coated paper only ... ... One day, The professor received a phone call from an economist who asked him: If there is a point P in the equilateral triangle, then no matter how the position of the point P changes in the triangle, the distance from the triangle P to the triangle And always is the same? Professor Peto immediately gave him a satisfactory answer .As shown in Figure 1, △ ABC is divided into △ PAB, △ PBC, △ PCA.AFBDCE Pz xy Figure 1 with x, y, z Respectively, remember P to △ ABC three sides of the distance. As △ ABC equal to the three sides, as a, then S △ PBC = 12ax, S △ PAC = 12ay, S △ PAB = 12az, then x + y + z = 2S △ ABC a①.