Vecten points

Vecten points

Outer Vecten point

Let ABC be any given plane triangle. On the sides BC, CA, AB of the triangle, construct outwardly drawn three squares with centres $O_{a},O_{b},O_{c}$ respectively. Then the lines ${\displaystyle AO_{a},BO_{b}}$ and $CO_{c}$ are concurrent. The point of concurrence outer is Vecten point of the triangle ABC.

In Clark Kimberling's Encyclopedia of Triangle Centers, the outer Vecten point is denoted by X(485).^[1] The Vecten points are named after an early 19th-century French mathematician named Vecten, who taught mathematics with Gergonne in Nîmes and published a study of the figure of three squares on the sides of a triangle in 1817.^[2]

Inner Vecten point

Let ABC be any given plane triangle. On the sides BC, CA, AB of the triangle, construct inwardly drawn three squares respectively with centres ${\displaystyle I_{a},I_{b},I_{c}}$ respectively. Then the lines ${\displaystyle AI_{a},BI_{b}}$ and ${\displaystyle CI_{c}}$ are concurrent. The point of concurrence is inner Vecten point of the triangle ABC.

In Clark Kimberling's Encyclopedia of Triangle Centers, the inner Vecten point is denoted by X(486).^[1]

The line ${\displaystyle X(485)X(486)}$ meets the Euler line at the Nine point center of the triangle $ABC$. The Vecten points lie on the Kiepert hyperbola

See also

• Napoleon points, a pair of triangle centers constructed in an analogous way using equilateral triangles instead of squares

References

1. Kimberling, Clark. "Encyclopedia of Triangle Centers".
2. Ayme, Jean-Louis, La Figure de Vecten (PDF), retrieved 2014-11-04.

External links

• Weisstein, Eric W. "Vecten Points". MathWorld.