## X(371) (KENMOTU POINT (CONGRUENT SQUARES POINT))

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

 The JRE (Java Runtime Environment) is not enabled in your browser!

This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           cos(A - π/4) : cos(B - π/4) : cos(C - π/4)
= cos A + sin A : cos B + sin B : cos C + sin C

Barycentrics    sin A cos(A - π/4) : sin B cos(B - π/4) : sin C cos(C - π/4)

There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X(371), may have first been published in Kenmotu's Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.

The edgelength of the three squares is 21/2abc/(a2 + b2 + c2 + 4σ), where σ = area(ABC). (Edward Brisse, 2/12/00)

X(371) is the internal center of similitude of the circumcircle and the 2nd Lemoine circle (cosine circle) (Peter J. C. Moses, 5/9/03). Also, X(371) is the internal center of similitude of the Gallatly circle (defined just before X(2007)) and the 1st Lemoine circle (Peter J. C. Moses, 9/10/03).

Hidetoshi Fukagawa, Traditional Japanese Mathematics Problems of the 18th and 19th Centuries, forthcoming.

Floor van Lamoen, Vierkanten in een driehoik: 3. Congruente vierkanten

Tony Rothman, with the cooperation of Hidetoshi Fukagawa, Japanese Temple Geometry (feature article in Scientific American)

X(371) lies on these lines:
2,486    3,6    4,485    25,493    140,615    193,488    315,491    492,641    601,606    602,605

X(371) is the {X(3),X(6)}-harmonic conjugate of X(372).

X(371) = reflection of X(I) in X(J) for these (I,J): (315,640), (372,32), (637,639)
X(371) = isogonal conjugate of X(485)
X(371) = inverse-in-Brocard-circle of X(372)
X(371) = inverse-in-1st-Lemoine-circle of X(2461)
X(371) = complement of X(637)
X(371) = anticomplement of X(639)
X(371) = X(4)-Ceva conjugate of X(372)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.