Inradius

Author

Eric W. Weisstein
May 11, 2003

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

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

Figure

tri=Triangle[{{0.,0.},{1.3,1.7},{3.,0.}}];
v=Vertices[tri];
incenter=Coordinates[Incenter[tri]];
r=Inradius[tri];
redpts={incenter,O,incenter+{r,0},X};
pts={v[[1]],A,v[[2]],B,v[[3]],C};

Show[Graphics[{
tri,
{
Red,
Incircle[tri],
Line[{incenter,incenter+{r,0}}],
PointLabels[redpts,.02]
},
PointLabels[pts,.02]
}], AspectRatio->Automatic,PlotRange->All]

Identities

tri=a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b;

r=((a b c)/(4 s R))

Resolve[ForAll[{a,b,c},tri,
R r==a b c/(4s)/.{r->Inradius[],R->Circumradius[],s->Semiperimeter[]}
]]//Timing

(Δ^2)=r (Subscript[r, 1]) (Subscript[r, 2]) (Subscript[r, 3])

Resolve[ForAll[{a,b,c},tri,
Area[]^2==r rs/.{r->Inradius[],rs->Times@@Exradii[]}
]]//Timing

cos(A)+cos(B)+cos(C)=(r/R)+1

Solve[Cos[A]+Cos[B]+Cos[C]==r/R+1,r]

Resolve[ForAll[{a,b,c},tri,
DistanceTrilinears[Cos[A]+Cos[B]+Cos[C]]==r/R+1/.{r->Inradius[],R->Circumradius[]}
]]//Timing