若三角形一边上的点和这边所对的顶点平分三角形的周界,则称这一点为三角形的周界中点,并将以三个周界中点为顶点的三角形称为周界中点三角形.图1如图1,D、E、F为△ABC的周界中点,设BC=a,AC=b,AB=c,s=12(a+b+c),EF=a1,DF=b1,DE=c1,△ABC的内切圆半径为r,外接圆半径为R.文[1]证 If the point on one side of the triangle and the vertex on this side equally divide the perimeter of the triangle, it is called the midpoint of the perimeter of the triangle, and the triangle with the midpoint of the three perimeters as the vertex is called the perimeter midpoint triangle. Figure 1 1, D, E, and F are the midpoints of the perimeter of △ABC, set BC=a, AC=b, AB=c, s=12(a+b+c), EF=a1, DF=b1, DE=c1, The radius of the inscribed circle of ABC is r, and the radius of the circumscribed circle is R.[1].