Isoperimetric point

File:Isoperimetric point.svg

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 {{mvar|P}} in the plane of a triangle {{math|△ABC}} having the property that the triangles {{math|△PBC, △PCA, △PAB}} have isoperimeters, that is, having the property that{{cite journal|last=G. R. Veldkamp|title=The isoperimetric point and the point(s) of equal detour|journal=Amer. Math. Monthly|year=1985|volume=92|issue=8|pages=546–558|doi=10.2307/2323159|jstor=2323159}}{{cite journal|last=Hajja|first=Mowaffaq|author2=Yff, Peter|title=The isoperimetric point and the point(s) of equal detour in a triangle|journal=Journal of Geometry|year=2007|volume=87|issue=1–2|pages=76–82|doi=10.1007/s00022-007-1906-y|s2cid=122898960}}

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

Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of {{math|△ABC}} in the sense of Veldkamp, if it exists, has the following trilinear coordinates.{{cite web|last=Kimberling|first=Clark|title=Isoperimetric Point and Equal Detour Point|url=http://faculty.evansville.edu/ck6/tcenters/recent/isoper.html|access-date=27 May 2012}}

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

Given any triangle {{math|△ABC}} one can associate with it a point {{mvar|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 {{math|△ABC}}. It is designated as the triangle center X(175). The point X(175) need not be an isoperimetric point of triangle {{math|△ABC}} in the sense of Veldkamp. However, if isoperimetric point of triangle {{math|△ABC}} in the sense of Veldkamp exists, then it would be identical to the point X(175).

The point {{mvar|P}} with the property that the triangles {{math|△PBC, △PCA, △PAB}} have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.{{cite web|last=Kimberling |first=Clark |title=X(175) Isoperimetric Point |url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |access-date=27 May 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |archive-date=19 April 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.[http://gallica.bnf.fr/ark:/12148/bpt6k201173h/f3.image Gallica]

Existence of isoperimetric point in the sense of Veldkamp

File:Isoperimetric point 02.svg

Let {{math|△ABC}} be any triangle. Let the sidelengths of this triangle be {{mvar|a, b, c}}. Let its circumradius be {{mvar|R}} and inradius be {{mvar|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 {{math|△ABC}} has an isoperimetric point in the sense of Veldkamp if and only if $a + b + c > 4R + r.$

For all acute angled triangles {{math|△ABC}} we have {{math|a + b + c > 4R + r}}, and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.

Properties

Let {{mvar|P}} denote the triangle center X(175) of triangle {{math|△ABC}}.

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

$\frac{2 \triangle}{\bigl| 4R + r - (a+b+c) \bigr|}$

where {{math|△}} is the area, {{mvar|R}} is the circumradius, {{mvar|r}} is the inradius, and {{mvar|a, b, c}} are the sidelengths of {{math|△ABC}}.

Soddy circles

File:Inner and outer Soddy points 02.svg

File:Inner and outer Soddy points 01.svg

Given a triangle {{math|△ABC}} one can draw circles in the plane of {{math|△ABC}} with centers at {{mvar|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 {{mvar|A, B, C}} as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of {{math|△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 {{math|△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 {{math|△ABC}}.

{{cite journal|last=Nikolaos Dergiades|title=The Soddy Circles|journal=Forum Geometricorum|year=2007|volume=7|pages=191–197|url=http://forumgeom.fau.edu/FG2007volume7/FG200726.pdf|access-date=29 May 2012|archive-date=14 June 2010|archive-url=https://web.archive.org/web/20100614151039/http://forumgeom.fau.edu/FG2007volume7/FG200726.pdf|url-status=dead}}{{cite web|title=Soddy Circles|url=http://oz.nthu.edu.tw/~g9721504/soddycircles.html|access-date=29 May 2012}}

The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of {{math|△ABC}}.

References

External links

[https://www.geogebra.org/m/p897we6j isoperimetric and equal detour points] - interactive illustration on Geogebratube

Category:Triangle centers