Taylor circle

The feet of the perpendiculars $P_{ab},P_{ac},P_{ba},P_{bc},P_{ca},P_{cb}$ lie on a common circle

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.^[1] However it was already considered by Eugène Charles Catalan in 1879 and first proposed by the French mathematician Eutaris in 1877.^[2]

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.^[3]

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:^[4]

r_{T}=R{\sqrt {\sin ^{2}\alpha \sin ^{2}\beta \sin ^{2}\gamma +\cos ^{2}\alpha \cos ^{2}\beta \cos ^{2}\gamma }}

References

^ Claudi Alsina, Roger B. Nelsen: Icons of Mathematics. An Exploration of Twenty Key Images. Springer, 2011, ISBN 99780883859865, pp. 112–113 (excerpt (Google), p. 112, at Google Books)
^ Taylor-Kreis at cut-the-knot.org (retrieved 2024-0-2-9)
^ Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, p. 277
^ Weisstein, Eric W. "Taylor Circle". MathWorld.

External links

Weisstein, Eric W. "Taylor Circle". MathWorld.
Taylor-Kreis at cut-the-knot.org

[1] Claudi Alsina, Roger B. Nelsen: Icons of Mathematics. An Exploration of Twenty Key Images. Springer, 2011, ISBN 99780883859865, pp. 112–113 (excerpt (Google), p. 112, at Google Books)

[2] Taylor-Kreis at cut-the-knot.org (retrieved 2024-0-2-9)

[3] Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, p. 277

[4] Weisstein, Eric W. "Taylor Circle". MathWorld.

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See also

References

External links