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Kenmotu Point


KenmotuPoint

A the (first, or internal) Kenmotu point, also called the congruent squares point, is the triangle center constructed by inscribing three equal squares such that each square touches two sides and all three squares touch at a single common point. The Kenmotu point is Kimberling center X_(371) and has equivalent triangle center functions

alpha_(371)=cos(A-1/4pi)
(1)
alpha_(371)=cosA+sinA.
(2)

It was first found by J. Rigby (Kimberling 1998, p. 268).

The contact points of the squares with the sides are concyclic and lie on the Kenmotu circle, which has radius equal to 1/sqrt(2) times the square edge length.

The definition in terms of inscribed squares breaks down for certain classes of triangles, at which point the vertices opposite the Kenmotu point can lie outside the triangle DeltaABC and the isosceles right triangles that remain in the interior of the triangle can overlap one another. This happens when the Kenmotu circle becomes tangent to a side.

The edge lengths of the inscribed squares are

 a^'=(sqrt(2)abc)/(a^2+b^2+c^2+4Delta),
(3)

where Delta is the area of the reference triangle DeltaABC (Kimberling 1998, p. 268).

The Kenmotu point lines on the Brocard axis.

Its isogonal conjugate is the inner Vecten point X_(485), and is the perspector of a number of pairs of named triangles.

The "free" vertices of the squares are perspective to the reference triangle at X_(372), which could be considered the second, or external, Kenmotu point.


See also

Ehrmann Congruent Squares Point, Kenmotu Circle, Sangaku Problem, Triangle Packing

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References

Danneels, E. "The Eppstein Centers and the Kenmotu Points." Forum Geom. 5, 173-180, 2005. http://forumgeom.fau.edu/FG2005volume5/FG200523index.html.Fukagawa, H. and Rigby, J. F. Traditional Japanese Mathematics Problems from the 18th and 19th Centuries. Singapore: Science Culture Technology Press, 2002.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Kenmotu Point

Cite this as:

Weisstein, Eric W. "Kenmotu Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KenmotuPoint.html

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