Isoperimetric point

Let be any triangle. Let the sidelengths of this triangle be . Let its circumradius be and inradius be . The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows. :The triangle has an isoperimetric point in the sense of Veldkamp if and only if a + b + c > 4R + r. For all acute angled triangles we have , and so all acute angled triangles have isoperimetric points in the sense of Veldkamp. ==Properties==

Let denote the triangle center X(175) of triangle . • lies on the line joining the incenter and the Gergonne point of . • If is an isoperimetric point of in the sense of Veldkamp, then the excircles of triangles are pairwise tangent to one another and is their radical center. • If is an isoperimetric point of in the sense of Veldkamp, then the perimeters of are equal to \frac{2 \triangle}{\bigl| 4R + r - (a+b+c) \bigr|} where is the area, is the circumradius, is the inradius, and are the sidelengths of . ==Soddy circles==

Given a triangle one can draw circles in the plane of with centers at such that they are tangent to each other externally. In general, one can draw two new circles such that each of them is tangential to the three circles with as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of . The circle with the smaller radius is the inner Soddy circle and its center is called the inner Soddy point or inner Soddy center of . The circle with the larger radius is the outer Soddy circle and its center is called the outer Soddy point or outer Soddy center of triangle . The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of . ==References==