Tripolar Coordinates

Author

Eric W. Weisstein
January 8, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/TripolarCoordinates.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TripolarCoordinates.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identity

Triangle[c,b,a]

Expand[(((X-((-(a^2)+(b^2)+(c^2))/(2 c))))^2)+(((Y-((Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])/(2 c))))^2)]-z^2

Simplify[(b^2)+(((a^2) X)/c)-(((b^2) X)/c)-c X+(X^2)+(Y^2)-(z^2)]

Factor/@GroebnerBasis[{
X^2+Y^2-x^2,
(X-c)^2+Y^2-y^2,
((((((b^2) (c-X))/c)+(((a^2) X)/c)-c X+(X^2)+(Y^2)-(z^2)))^2)-(((a+b-c) (a-b+c) (-a+b+c) (a+b+c))/(c^2))(Y^2)
},{x,y,z},{X,Y},MonomialOrder->EliminationOrder]

eqn=%[[1]];

Factor[eqn+(((a^2)+(b^2)-(c^2))(x^2y^2+c^2z^2)+(a^2-b^2+c^2)(x^2z^2+b^2y^2)+(-a^2+b^2+c^2)(y^2z^2+a^2x^2))-(a^2x^4+b^2y^4+c^2z^4)-a^2b^2c^2]

Special Points

Centroid

FullSimplify[Together//@ToSidelengths[Distance[Trilinears[Centroid],#]&/@IdentityMatrix[3]],TriangleQ[]]

Circumcenter

Refine[Together//@ToSidelengths[Distance[Trilinears[Circumcenter],#]&/@IdentityMatrix[3]],TriangleQ[]]/.((a b c)/(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])):>Circumradius

Gergonne Point

FullSimplify[Together//@ToSidelengths[Distance[Trilinears[GergonnePoint],#]&/@IdentityMatrix[3]],TriangleQ[]]

Refine[Map[Factor,%,3],TriangleQ[]]

Incenter

FullSimplify[Together//@ToSidelengths[Distance[Trilinears[Incenter],#]&/@IdentityMatrix[3]],TriangleQ[]]

Nagel Point

Refine[Map[Factor,Together//@ToSidelengths[Distance[Trilinears[NagelPoint],#]&/@IdentityMatrix[3]],3],TriangleQ[]]

Nine-Point Center

Factor[Together//@ToSidelengths[Distance[Trilinears[NinePointCenter],#]&/@IdentityMatrix[3]/.Cos[x_]:>TrigExpand[Cos[x]]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

Map[Factor,%/.(1/(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])):>(R/(a b c)),3]

ToAngles[%]

Orthocenter

Simplify[ToAngles[FullSimplify[Together//@ToSidelengths[Distance[Trilinears[Orthocenter],#]&/@IdentityMatrix[3]],TriangleQ[]]]/.(1/(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])):>(Circumradius/(a b c)),TriangleQ[]]

Symmedian Point

FullSimplify[Together//@ToSidelengths[Distance[Trilinears[SymmedianPoint],#]&/@IdentityMatrix[3]],TriangleQ[]]