Vecten point

First Vecten point V ₁ with Kiepert triangle (red)

{\ displaystyle {\ mathcal {K}} _ {\ frac {\ pi} {4}}}

The two Vecten points belong to the special points of a triangle . They go back to the French mathematician Vecten (1811/1812).

First Vecten point

Three squares are drawn outwards over the sides of a triangle ABC . Each of the three centers of the square that make up the Kiepert triangle is connected to the opposite corner of the original triangle. The connecting straight lines intersect at a point which is called the first Vecten point and bears the Kimberling number X (485). ${\ displaystyle {\ mathcal {K}} _ {\ frac {\ pi} {4}}}$

Second Vecten point

Second Vecten point V ₂

If you draw the squares inwards instead of outwards, you get the second Vecten point with Kimberling number X (486).

properties

The two Vecten points lie on the Kiepert hyperbola .
The Vecten points lie on a straight line with the center of the Feuerbach circle (nine-point circle).

Coordinates

Vecten points ( and ) ${\ displaystyle X_ {485}}$ ${\ displaystyle X_ {486}}$
Trilinear coordinates	${\ displaystyle \ sec \ left (\ alpha \ pm {\ frac {\ pi} {4}} \ right) \,: \, \ sec \ left (\ beta \ pm {\ frac {\ pi} {4} } \ right) \,: \, \ sec \ left (\ gamma \ pm {\ frac {\ pi} {4}} \ right)}$
Barycentric coordinates	${\ displaystyle \ sin \ alpha \ cdot \ sec \ left (\ alpha \ pm {\ frac {\ pi} {4}} \ right) \,: \, \ sin \ beta \ cdot \ sec \ left (\ beta \ pm {\ frac {\ pi} {4}} \ right) \,: \, \ sin \ gamma \ cdot \ sec \ left (\ gamma \ pm {\ frac {\ pi} {4}} \ right) }$
The plus sign applies to the first Vecten point, the minus sign to the second.

literature

Sotirios E. Louridas, Michael Th. Rassias: Problem-Solving and Selected Topics in Euclidean Geometry: In the Spirit of the Mathematical Olympiads . Springer, 2014, ISBN 978-1-4614-7273-5 , pp. 62-63
Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , pp. 4-7, 93

Web links

Kimberling's Encyclopedia of Triangle Centers (English)
Eric W. Weisstein : Vecten Points . In: MathWorld (English).

Individual evidence

↑ Little is known about Vecten. It is known that from 1810 to 1816 he worked as a " professeur de mathématiques spéciales " at the Lycée de Nîmes and published 22 articles in the journal Annales published by the mathematician Joseph Gergonne . See: Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , p. 4

[1] Little is known about Vecten. It is known that from 1810 to 1816 he worked as a " professeur de mathématiques spéciales " at the Lycée de Nîmes and published 22 articles in the journal Annales published by the mathematician Joseph Gergonne . See: Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , p. 4