Taylor circle

Graphic representation of the Taylor circle

The Taylor circle of a triangle is one of the special circles in triangle geometry .

The Taylor circle is named after Henry Martin Taylor (1842–1927).

To get the Taylor circle, you first have to draw the heights , i.e. the perpendiculars from the corners of the triangle to the opposite sides. Then two perpendiculars are felled from each of the three height base points on the two neighboring sides. These six perpendiculars are also known as the minor heights of the triangle. It can be proven that the six base points of the minor heights lie on a circle. In order to find the center of this so-called Taylor circle, one only needs to bring two perpendicular lines for each two of the six foot points mentioned to the intersection.

radius

The radius of the Taylor circle is

${\displaystyle R_{T}=R{\sqrt {\sin ^{2}\alpha \sin ^{2}\beta \sin ^{2}\gamma +\cos ^{2}\alpha \cos ^{2}\beta \cos ^{2}\gamma }},}$

where R is the perimeter radius .

Focus

The center of the Taylor circle has the barycentric coordinates

${\displaystyle {\Big (}\sin \alpha [\cos \alpha -\cos(2\alpha )\cos(\beta -\gamma )]:\sin \beta [\cos \beta -\cos(2\beta )\cos(\gamma -\alpha )]:\sin \gamma [\cos \gamma -\cos(2\gamma )\cos(\alpha -\beta )]{\Big )}}$

and lies on the Brocard axis . He's got the kimberling number ${\displaystyle X(389).}$

Web links

• Eric W. Weisstein : Taylor Circle . In: MathWorld (English).
• Taylor circle - a visualization with GeoGebra

See also

• Excellent points in the triangle
• Circles on the triangle