Isoperimetric point

In geometry, the isoperimetric point is a triangle center — a special point associated with a plane triangle.

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, △PAB have isoperimeters, that is, having the property that[1][2]

{\displaystyle {\begin{aligned}&{\overline {PB}}+{\overline {BC}}+{\overline {CP}},\\=\ &{\overline {PC}}+{\overline {CA}}+{\overline {AP}},\\=\ &{\overline {PA}}+{\overline {AB}}+{\overline {BP}}.\end{aligned}}}

Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions.

The isoperimetric point of △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.

However, if isoperimetric point of triangle △ABC in the sense of Veldkamp exists, then it would be identical to the point X(175).

The point P with the property that the triangles △PBC, △PCA, △PAB have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.

Let the sidelengths of this triangle be a, b, 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.

[1] 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.

where △ is the area, R is the circumradius, r is the inradius, and a, b, c are the sidelengths of △ABC.

[6] Given a triangle △ABC one can draw circles in the plane of △ABC with centers at A, B, C such that they are tangent to each other externally.

(One of the circles may degenerate into a straight line.)

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 △ABC.

[6][7] The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of △ABC.

A triangle △ ABC in which the triangle center X (175) is not the isoperimetric point in the sense of Veldkamp.
Inner and outer Soddy circles in the case where the outer Soddy point is an isoperimetric point in the sense of Veldkamp.
Inner and outer Soddy circles in the case where the outer Soddy point is not an isoperimetric point in the sense of Veldkamp.