Triangle Arcs

Author

Eric W. Weisstein
May 10, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Distances

soln=Solve[{x+y==c,y+z==b,x+y+z==a},{x,y,z}]

Simplify[Subtract@@({y^2,2x z}/.soln[[1]])]

eqn={y^2,2x z}/.soln[[1]]/.a->(Sqrt[(b^2)+(c^2)])

Simplify[Subtract@@eqn]

Area

Solve[{x^2+y^2==c^2&&(x-a)^2+y^2==b^2},{x,y}]

With[{a=1,b=.5},With[{c=(Sqrt[(a^2)-(b^2)])},
Module[{
x=(a^2-b^2+c^2)/(2a),
y
},
y=(Sqrt[(c^2)-(x^2)]);
Show[{
Graphics[Line[{{0,0},{a,0},{x,y},{0,0}}]],
RegionPlot[(x-a)^2+y^2<b^2&&x^2+y^2<c^2&&y>0,{x,-1,1},{y,-1,1},PlotPoints->100]
}]
]]]

area=FullSimplify[Integrate[Boole[(x-a)^2+y^2<b^2&&x^2+y^2<c^2&&y>0/.c->(Sqrt[(a^2)-(b^2)])],
{x,-∞,∞},{y,-∞,∞},Assumptions->0<b<a],0<b<a]

Simplify[area/.(a-b)(a+b):>c^2/.a^2-b^2:>c^2,c>0]//Factor