Carnot's Theorem

Author

Eric W. Weisstein
August 25, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{Blue,pedals=Cartesian[#[t]]&/@PedalLines[Circumcenter],
PerpSquare[#,.03,1]&/@Transpose[{Sides[t],pedals}]},
{Red,Incircle[t],Circumcircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Cartesian[#[t]]&/@Vertices[PedalTriangle[Circumcenter]],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"},Thread[LabelOffset->.04{1,{-1.2,.05},{0,-1}}]}],
{
{Circumcenter[t],"O",Red},
{Incenter[t],"I",Red}
}
],
LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Computation

d=Factor/@Together//@ToSidelengths[Distance@@@PedalLines[Circumcenter]]

sum=Simplify[Together//@(Plus@@d),TriangleQ[]]

Simplify[ToAngles[sum],TriangleQ[]&&Area∈Reals]

Factor[Simplify[sum,AcuteQ[]]]

Simplify[{(a^2)+(b^2)-(c^2)>0,(a^2)-(b^2)+(c^2)>0,-(a^2)+(b^2)+(c^2)>0},AcuteQ[]]

Factor[Simplify[ToSidelengths[Circumradius+Inradius],TriangleQ[]]]