Right Triangle

Author

Eric W. Weisstein
July 31, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=y≥0&&a b≥b x+a y&&x≥0;

implreg=ImplicitRegion[ineq,{x,y}];

verts={{0,0},{a,0},{0,b}};

reg=Triangle[verts]

assum=a>0&&b>0;

t=FullSimplify[Triangle[𝒶,𝒷,Sqrt[(𝒶^2)+(𝒷^2)]],𝒶>0&&𝒷>0]

abc=Simplify[SideLengths[Evaluate[t]],𝒶>0&&𝒷>0]

Plots

Diagram

LaminaData["FilledRightTriangle","Diagram"]

DiscretizeRegion

Block[{a=1,b=.5},DiscretizeRegion[#]]&/@{reg,implreg}

RegionPlot

Block[{a=1,b=2},RegionPlot[ineq,{x,0,a},{y,0,b}]]

Schematic Figure

right=Triangle[{{0,0},{b,0},{0,a}}];

r=right/.{a->1,b->1.5};
pts={v[[1]],C,v[[2]],A,v[[3]],B};

Show[Graphics[{r,
PerpSquare[Rest[Sides[r]],.05,4],
PointLabels[r,pts,.03,-.05]
},AspectRatio->Automatic]
]

Sangaku

Show[Graphics[RightTriangleSquares[8,{3,1.6}]],AspectRatio->Automatic,PlotRange->All]

rr[3,{a,b}]^2-rr[1,{a,b}]rr[5,{a,b}]

Distinct right triangles having identical areas

l1={{1380,19019,19069},{3059,8580,9109},{4485,5852,7373}};

RightQ[SidesToTriangle@@#]&/@l1

TriangleArea[Triangle[#]]&/@{{1380,19019,19069},{3059,8580,9109},{4485,5852,7373}}

l2={{24,70,74},{40,42,58},{15,112,113}};

RightQ[SidesToTriangle@@#]&/@l2

TriangleArea[Triangle[#]]&/@l2

Properties

Area

Area

Assuming[assum,FullSimplify[Area[reg]]]//Timing

Assuming[assum,FullSimplify[Area[implreg]]]//Timing

Formula

TriangleArea[right]

Integrate

Assuming[a>0&&b>0,Area[ineq[x,y],{x,y}]]

RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[reg]]]

Assuming[assum,FullSimplify[RegionMeasure[implreg]]]

AreaInertiaTensor

Assuming[a>0&&b>0,AreaInertiaTensor[ineq[x,y],{x,y}]]

Circumradius

FullSimplify[(ToSidelengths[Circumradius]/.Thread[{a,b,c}->abc]),𝒶>0&&𝒷>0]

Centroid

Integrate

Assuming[a>0&&b>0,Centroid[ineq[x,y],{x,y}]]

RegionCentroid

Assuming[assum,FullSimplify[RegionCentroid[reg]]]//Timing

Assuming[assum,FullSimplify[RegionCentroid[implreg]]]//Timing

GeneralizedDiameter

ineqs=FullSimplify[And@@ineq@@@{{x1,y1},{x2,y2}},a>0&&b>0]

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2},Reals]//Timing

Inradius

FullSimplify[(ToSidelengths[Inradius]/.Thread[{a,b,c}->abc]),𝒶>0&&𝒷>0]

((Sqrt[((𝒶 𝒷 (𝒶+𝒷-(Sqrt[(𝒶^2)+(𝒷^2)])))/(𝒶+𝒷+(Sqrt[(𝒶^2)+(𝒷^2)])))])/(Sqrt[2]))/.{𝒶->a,𝒷->b}

RadiiOfGyration

Assuming[a>0&&b>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]

Inradius

r=right/.{a->1,b->1.5};
i=Incenter[r];
v=Vertices[r];

pts={v[[1]],C,v[[2]],A,v[[3]],B};
bluepts={i,"\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)"};
Show[Graphics[{
r,
PerpSquare[Rest[Sides[r]],.05,4],
{Thickness[.01],Incircle[r]},
{
Red,
PerpSquare[#,.05,2]&/@Transpose[{PedalLines[r,i],Sides[r]}],Dashing[{.02,.02}],
PedalLines[r,i]
},
{
Blue,
PointLabels[r,bluepts,.03,.05],Line[{#[[1]],Coordinates@i,#[[2]]}]&/@Partition[Vertices[r],2,1,1]
},
PointLabels[r,pts,.03,-.05]
},AspectRatio->Automatic]
]

i=PowerExpand[Inradius[right]]

i/.Sqrt[a^2+b^2]:>c

FullSimplify[i^2]

FullSimplify[(c^2+a b-(a+b)c)/2]

FullSimplify[((c-a)(c-b)/2)-((a+b-c)/2)^2/.c->Sqrt[a^2+b^2]]

FullSimplify[((c-a)(c-b)/2)-(a b/(a+b+c))^2/.c->Sqrt[a^2+b^2]]

r[a_,b_]:=Module[{c=Sqrt[a^2+b^2]},
{Inradius[Triangle[{{0,0},{0,a},{b,0}}]],Sqrt[(c-a)(c-b)/2],(a+b-c)/2,a b/(a+b +c)}
]

{r[3,4],r[5,12],r[8,15],r[7,24]}//ColumnForm

Circumradius

FullSimplify[PowerExpand[Circumradius[Triangle[{{0,0},{b,0},{0,a}}]]]]

Sqrt[Simplify[%^2]]

FullSimplify[a b c/Sqrt[(a+b+c)(b+c-a)(c+a-b)(a+b-c)]/.c->Sqrt[a^2+b^2]]

PowerExpand[%]

Median and Altitude

t=Triangle[5,4,3];

Show[Graphics[{
t,
{Blue,
AngleBisectors[t][[1]],
EqualAngleDivisionTicks[t,2,.4,.1,.4,VertexArcs->True][[1]]
},
{Green,Medians[t][[1]]},
{Red,Altitudes[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Midpoints[t][[1]],"(Subscript[M, A])", Green},
{Feet[Altitudes[t]][[1]],"(Subscript[H, A])", Red},
{EndPoint[AngleBisectors[t][[1]]],"(Subscript[T, A])",Blue}
}
],PointSize[.04],LabelOffset->-.2]
}],AspectRatio->Automatic,PlotRange->All]