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Dou Circle


DouCircle

The Dou circle is the circle cutting the sidelines of the reference triangle DeltaABC at A^', A^(''), B^', B^(''), C^', and C^('') such that ∠A^'AA^('')=∠B^'BB^('')=∠C^'CC^('')=90 degrees (i.e., all are right angles).

Its center is Kimberling center X_(155) with center function

 alpha_(155)=cosA(-cos^2A+cos^2B+cos^2C)
(1)

and its radius is

 R_D=(sqrt(f(a,b,c))R)/(8a^3b^3c^3|cosAcosBcosC|),
(2)

where R is the circumradius of the reference triangle and f(a,b,c) is an 18th-degree polynomial.

It has circle function

 l=-(a^2(S_A^2-S_BS_C)-S_A(S_B^2+S_C^2))/(2bcS_BS_C),
(3)

which does not correspond to any Kimberling center.

No Kimberling centers lie on the Dou circle.


See also

Central Circle, Perpendicular

Explore with Wolfram|Alpha

References

Dou, J. "Problem 1140." Crux Math. 12, 79-80, 1986.Dou, J. "Problem 1140." Crux Math. 28, 461-462, 2002.Sokolowsky, D. "Problem 1140." Crux Math. 13, 232-235, 1987.

SeeAlso

Central Circle

Referenced on Wolfram|Alpha

Dou Circle

Cite this as:

Weisstein, Eric W. "Dou Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DouCircle.html

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