Isosceles Triangle

Author

Eric W. Weisstein
July 23, 2014

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

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

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

Definitions

ineq=-a/2≤x≤a/2&&0≤y≤(1-Abs[x/(a/2)]) h;

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

reg=Triangle[{{-a/2,0},{a/2,0},{0,h}}];

assum=a>0&&h>0;

t=Triangle[{{-𝒶/2,0},{𝒶/2,0},{0,h}}]

Figure

With[{a=1,h=1.3},Graphics[{
Line[v={{-a/2,0},{a/2,0},{0,h},{-a/2,0}}],
{Dashing[.04],Line[{{0,0},{0,h}}]},
EqualAngleTicks[v[[#]],1,.15,.04,1]&/@{{3,1,2},{1,2,3}},
Text[Style["a",Italic],{0,-.05}],
Text[Style["b",Italic],#]&/@{{-.33,.65},{.33,.65}},
Text[Style["h",Italic],{.05,h/2-.1}]
}]]

Plots

Diagram

LaminaData["FilledIsoscelesTriangle","Diagram"]

DiscretizeRegion

Block[{a=1, h=1},Show[DiscretizeRegion[reg],ImageSize->Small]]

RegionPlot

Block[{a=1,h=1},RegionPlot[ineq,{x,-a/2,a/2},{y,0,h},AspectRatio->Automatic,ImageSize->Small,Frame->False]]

InequalityPlot

<<Graphics`InequalityGraphics`

With[{a=1,h=2},InequalityPlot[-a/2<x<a/2&&0<y<(1-Abs[(x/(a/2))])h,{x},{y},Ticks->None]]

Equations

Inequality

ineq[x_,y_]:=-a/2<x<a/2&&0<y<(1-Abs[(x/(a/2))])h

Block[{a=1,h=2},
Show[{
RegionPlot[ineq[x,y],{x,-a/2,a/2},{y,0,h},AspectRatio->Automatic],
Graphics[{Blue,inc,Red,circumc}]
}]
]

Properties

Area

Area

Assuming[assum,Simplify@Area[reg]]

Integrate

Integrate[Boole[-a/2<x<a/2&&0<y<(1-Abs[(x/(a/2))])h],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&h>0]

Assuming[a>0&&h>0,Area[-a/2≤x≤a/2&&0≤y≤(1-Abs[x/(a/2)]) h,{x,y}]]

Integrate[Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&h>0]

RegionMeasure

Assuming[assum,Simplify@RegionMeasure[reg]]

Centroid

Integrate

Assuming[a>0&&h>0,Centroid[-a/2≤x≤a/2&&0≤y≤(1-Abs[x/(a/2)]) h,{x,y}]]

Integrate[{x,y}Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&h>0]

%/((a h)/2)

RegionCentroid

Assuming[assum,Simplify@RegionCentroid[reg]]

SideLengths

abc=Simplify[SideLengths[Evaluate[t]],𝒶>0&&h>0]/.𝒶->a

Incircle

inc=FullSimplify[Incircle[t],𝒶>0&&h>0]/.𝒶->a

Circumcircle

circumc=FullSimplify[Circumcircle[t],𝒶>0&&h>0]/.𝒶->a

InertiaTensor

Integrate[AreaInertiaTensor[{x,y}]Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&h>0]

RadiiOfGyration

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

Erecting Isosceles Triangles on Arbitrary Triangle

tri={{0,0},{1,0},{.4,.7}};

Show[Graphics[{
{Thickness[.012],Triangle[tri]},
RGBColor[0,0,1],e=ErectScaleneTriangles[tri,.5],
RGBColor[1,0,0],
Line/@(l=Transpose[{v=Last@@@e,tri}]),
PointSize[.03],Point[LineLineIntersection@@Take[l,2]],
RGBColor[0,0,1],PointLabels[Flatten[Transpose[{v,{"A'","B'","C'"}}],1],.03,.04],
Dashing[{.02,.02}],Triangle[v],
RGBColor[0,0,0],
PointLabels[Flatten[Transpose[{tri,{"A","B","C"}}],1],.03,.04]
}/.Triangle:>polygon,AspectRatio->Automatic],PlotRange->All]

PerspectiveQ[tri,v]

CollinearQ@PerspectiveAxis[tri,v]