Exradius

Author

Eric W. Weisstein
January 25, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

s=Plus@@Table[a[i],{i,3}]/2;
A=Sqrt[s Times@@Table[s-a[i],{i,3}]];
r=Sqrt[Times@@Table[s-a[i],{i,3}]/s];
R=Times@@Table[a[i],{i,3}]/(4s r);
rr[k_]:=A/(s-a[k])

triangleCond=a[1]>0&&a[2]>0&&a[3]>0&&a[1]+a[2]>a[3]&&a[1]+a[3]>a[2]&&a[2]+a[3]>a[1];

IsTrueQ[eqn_]:=Resolve[ForAll[{a[1],a[2],a[3]},triangleCond,eqn],{a[1],a[2],a[3]}]

Identities

IsTrueQ[4R==rr[1]+rr[2]+rr[3]-r]//Timing

IsTrueQ[1/rr[1]+1/rr[2]+1/rr[3]==1/r]//Timing

IsTrueQ[r rr[1] rr[2] rr[3]==A^2]//Timing

IsTrueQ[rr[2]rr[3]+rr[3]rr[1]+rr[1]rr[3]==s^2]//Timing

IsTrueQ[r(rr[2]rr[3]+rr[3]rr[1]+rr[1]rr[2])==s A]//Timing

IsTrueQ[r(rr[2]rr[3]+rr[3]rr[1]+rr[1]rr[2])==rr[1]rr[2]rr[3]]//Timing

IsTrueQ[r(rr[2]rr[3]+rr[3]rr[1]+rr[1]rr[2])==rr[1]rr[2]rr[3]]//Timing

IsTrueQ[r(rr[1]+rr[2]+rr[3])==a[2]a[3]+a[3]a[1]+a[1]a[2]-s^2]//Timing

IsTrueQ[r rr[1]+r rr[2]+r rr[3]+rr[1]rr[2]+rr[2]rr[3]+rr[3]rr[1]==a[2]a[3]+a[3]a[1]+a[1]a[2]]//Timing

IsTrueQ[rr[2]rr[3]+rr[3]rr[1]+rr[1]rr[2]-r rr[1]-r rr[2]-r rr[3]==1/2(a[1]^2+a[2]^2+a[3]^2)]//Timing

Check

t=Triangle[{{0,0},{1,0},{.4,.6}}];

((ToSidelengths[(Area/(Semiperimeter-a))])^2)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[((Semiperimeter(Semiperimeter-b)(Semiperimeter-c))/(Semiperimeter-a))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

((ToSidelengths[4Circumradius Sin[A/2]Cos[B/2]Cos[C/2]])^2)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[Evaluate[t]]]