Isoperimetric point

In geometry, the isoperimetric point is a triangle center — 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 Template:Mvar in the plane of a triangle Template:Math having the property that the triangles Template:Math have isoperimeters, that is, having the property that

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

Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of Template:Math in the sense of Veldkamp, if it exists, has the following trilinear coordinates.

$$\sec \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}-1:\sec \frac{B}{2}\cos \frac{C}{2}\cos \frac{A}{2}-1:\sec \frac{C}{2}\cos \frac{A}{2}\cos \frac{B}{2}-1$$

Given any triangle Template:Math one can associate with it a point Template:Mvar having trilinear coordinates as given above. This point is a triangle center and in Clark Kimberling's Encyclopedia of Triangle Centers (ETC) it is called the isoperimetric point of the triangle Template:Math. It is designated as the triangle center X(175). The point X(175) need not be an isoperimetric point of triangle Template:Math in the sense of Veldkamp. However, if isoperimetric point of triangle Template:Math in the sense of Veldkamp exists, then it would be identical to the point X(175).

The point Template:Mvar with the property that the triangles Template:Math have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.

Existence of isoperimetric point in the sense of Veldkamp

A triangle Template:Math in which the triangle center X(175) is not the isoperimetric point in the sense of Veldkamp.

Let Template:Math be any triangle. Let the sidelengths of this triangle be Template:Mvar. Let its circumradius be Template:Mvar and inradius be Template:Mvar. The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows.

The triangle Template:Math has an isoperimetric point in the sense of Veldkamp if and only if $$a+b+c>4R+r.$$

For all acute angled triangles Template:Math we have Template:Math, and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.

Properties

Let Template:Mvar denote the triangle center X(175) of triangle Template:Math.

• Template:Mvar lies on the line joining the incenter and the Gergonne point of Template:Math.
• If Template:Mvar is an isoperimetric point of Template:Math in the sense of Veldkamp, then the excircles of triangles Template:Math are pairwise tangent to one another and Template:Mvar is their radical center.
• If Template:Mvar is an isoperimetric point of Template:Math in the sense of Veldkamp, then the perimeters of Template:Math are equal to

$$\frac{2\mathrm{△}}{|4R+r-(a+b+c)|}$$ where Template:Math is the area, Template:Mvar is the circumradius, Template:Mvar is the inradius, and Template:Mvar are the sidelengths of Template:Math.

Soddy circles

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.

Given a triangle Template:Math one can draw circles in the plane of Template:Math with centers at Template:Mvar 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 Template:Mvar as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of Template:Math. 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 Template:Math. 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 Template:Math.

The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of Template:Math.

External links

• isoperimetric and equal detour points - interactive illustration on Geogebratube