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 $P$ 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]

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

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.^[3]

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

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 (ETC) it is called the isoperimetric point of the triangle $△ ABC$ . It is designated as the triangle center X(175).^[4] The point X(175) 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(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.^[4]^[5]

Existence of isoperimetric point in the sense of Veldkamp

Let $△ ABC$ be any triangle. 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]

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 $a + b + c > 4 R + r$ , and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.

Properties

Let $P$ denote the triangle center X(175) of triangle $△ ABC$ .^[4]

$P$ lies on the line joining the incenter and the Gergonne point of $△ ABC$ .
If $P$ is an isoperimetric point of $△ ABC$ in the sense of Veldkamp, then the excircles of triangles $△ PBC, △ PCA, △ PAB$ are pairwise tangent to one another and $P$ is their radical center.
If $P$ is an isoperimetric point of $△ ABC$ in the sense of Veldkamp, then the perimeters of $△ PBC, △ PCA, △ PAB$ are equal to

${\frac {2\triangle }{{\bigl |}4R+r-(a+b+c){\bigr |}}}$ where $△$ is the area, $R$ is the circumradius, $r$ is the inradius, and $a, b, c$ are the sidelengths of $△ ABC$ .^[6]

Soddy circles

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. In general, one can draw two new circles such that each of them is tangential to the three circles with $A, B, C$ as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of $△ ABC$ . 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 $△ ABC$ . 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 $△ ABC$ .

References

^ ^a ^b G. R. Veldkamp (1985). "The isoperimetric point and the point(s) of equal detour". Amer. Math. Monthly. 92 (8): 546–558. doi:10.2307/2323159. JSTOR 2323159.
^ Hajja, Mowaffaq; Yff, Peter (2007). "The isoperimetric point and the point(s) of equal detour in a triangle". Journal of Geometry. 87 (1–2): 76–82. doi:10.1007/s00022-007-1906-y. S2CID 122898960.
^ Kimberling, Clark. "Isoperimetric Point and Equal Detour Point". Retrieved 27 May 2012.
^ ^a ^b ^c Kimberling, Clark. "X(175) Isoperimetric Point". Archived from the original on 19 April 2012. Retrieved 27 May 2012.
^ The article by Emile Lemoine can be accessed in Gallica. The paper begins at page 111 and the point is discussed in page 126.Gallica
^ ^a ^b Nikolaos Dergiades (2007). "The Soddy Circles" (PDF). Forum Geometricorum. 7: 191–197. Retrieved 29 May 2012.
^ "Soddy Circles". Retrieved 29 May 2012.

External links

isoperimetric and equal detour points - interactive illustration on Geogebratube

[Veldkamp-1] G. R. Veldkamp (1985). "The isoperimetric point and the point(s) of equal detour". Amer. Math. Monthly. 92 (8): 546–558. doi:10.2307/2323159. JSTOR 2323159.

[2] Hajja, Mowaffaq; Yff, Peter (2007). "The isoperimetric point and the point(s) of equal detour in a triangle". Journal of Geometry. 87 (1–2): 76–82. doi:10.1007/s00022-007-1906-y. S2CID 122898960.

[3] Kimberling, Clark. "Isoperimetric Point and Equal Detour Point". Retrieved 27 May 2012.

[ETC-4] Kimberling, Clark. "X(175) Isoperimetric Point". Archived from the original on 19 April 2012. Retrieved 27 May 2012.

[5] The article by Emile Lemoine can be accessed in Gallica. The paper begins at page 111 and the point is discussed in page 126.Gallica

[Dergi-6] Nikolaos Dergiades (2007). "The Soddy Circles" (PDF). Forum Geometricorum. 7: 191–197. Retrieved 29 May 2012.

[7] "Soddy Circles". Retrieved 29 May 2012.

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