Taylor circle

{{Short description|Special circle in geometry}}

File:Taylor-Kreis.svg

The Taylor circle is a special circle associated with a triangle. It is named after Henry Martyn Taylor (1842–1927), who discussed it in 1882.Claudi Alsina, Roger B. Nelsen: Icons of Mathematics. An Exploration of Twenty Key Images. Springer, 2011, ISBN 99780883859865, pp. 112–113 ({{Google books|jl_0DwAAQBAJ|excerpt (Google)|page=112}}) However it was already considered by Eugène Charles Catalan in 1879 and first proposed by the French mathematician Eutaris in 1877.[https://www.cut-the-knot.org/triangle/Taylor.shtml Taylor-Kreis] at cut-the-knot.org (retrieved 2024-0-2-9)

Consider the three feet of the three altitudes of triangle. For each foot of an altitude construct the perpendiculars onto the two other triangles sides. The intersections of those perpendiculars with the triangles sides yield six points and those six points are located on a common circle, the Taylor circle.Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, p. 277

The radius of the Taylor circle can be computed by the following formula taking the three angles of a triangle and the radius of its circumcircle:{{MathWorld|TaylorCircle|Taylor Circle}}

: $r\_T\; =\; R\backslash sqrt\{\backslash sin^2\backslash alpha\backslash sin^2\backslash beta\backslash sin^2\backslash gamma+\backslash cos^2\backslash alpha\backslash cos^2\backslash beta\backslash cos^2\backslash gamma\}$