page_title: Isoperimetric point
section_title: Existence of isoperimetric point in the sense of Veldkamp
hierarchical_section_title: Isoperimetric point / Existence of isoperimetric point in the sense of Veldkamp
context_page_description: In geometry, the isoperimetric point is a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point P in the plane of a triangle ABC having the property that the triangles PBC, PCA and PAB have isoperimeters, that is, having the property that
  PB + BC + CP = PC + CA + AP = PA + AB + BP.
  Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of triangle ABC in the sense of Veldkamp, if it exists, has the following trilinear coordinates.
  Given any triangle ABC one can associate with it a point P having trilinear coordinates as given above. This point is a triangle center and in Clark Kimberling's Encyclopedia of Triangle Centers it is called the isoperimetric point of the triangle ABC. It is designated as the triangle center X. The point X need not be an isoperimetric point of triangle ABC in the sense of Veldkamp. However, if isoperimetric point of triangle ABC in the sense of Veldkamp exists, then it would be identical to the point X.
context_section_description: Let ABC be any triangle. Let the sidelengths of this triangle be a, b, and c. Let its circumradius be R and inradius be r. The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows.
  The triangle ABC has an isoperimetric point in the sense of Veldkamp if and only if a + b + c > 4R + r.
  For all acute angled triangles ABC we have a + b + c > 4R + r, and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.
caption: A triangle ABC in which the triangle center X(175) is not the isoperimetric point in the sense of Veldkamp.