Midpoint

Author

Eric W. Weisstein
March 1, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Trilinears

k[α_,β_,γ_]:=2Δ/(a α+b β+c γ)
x[α_,β_,γ_]:=((k[α,β,γ]β-r+(k[α,β,γ]α-r)Cos[C])/(Sin[C]))
y[α_,β_,γ_]:=k[α,β,γ]α-r
rr[α_,β_,γ_]:={x[α,β,γ],y[α,β,γ]}

m=FullSimplify[(rr[α2,β2,γ2]+rr[α1,β1,γ1])/2]

s=FullSimplify[Solve[{x==((k[α,β,γ]β-r+(k[α,β,γ]α-r)Cos[C])/(Sin[C])),y==k[α,β,γ]α-r},{α,β}]]

FullSimplify[s/. Thread[{x,y}->m]]

Check

<<PlaneGeometry`

t=Triangle[{{0,0},{1,0},{.2,.30}}];

FullSimplify[Midpoint[{Centroid,Orthocenter}]]

Midpoint[t,{Centroid,Orthocenter}]

Midpoint[Line[Coordinates/@{Centroid[t],Orthocenter[t]}]]

Midpoints of Triangle Sides

Midpoint/@Partition[IdentityMatrix[3],2,1,1]

KSection[#,2]&/@Partition[IdentityMatrix[3],2,1,1]

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Midpoints[t],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}]
],
{PointSize[.025],LabelOffset->.03}
],
EqualSideTicksAboutMidpoints[t,.02,.02]
}],AspectRatio->Automatic,PlotRange->All]

Triangle Midpoint Equidistance Offset Theorem

t=Triangle[{{0,0},{1,0},{.2,.7}}];
v={.08,.1,.12};
tp=Triangle[Coordinates/@Midpoints[t]+v SideUnitVectors[t]];
tq=Triangle[Coordinates/@Midpoints[t]-v SideUnitVectors[t]];
Show[Graphics[{
t,
{Red,tp},
{Blue,tq},
PointLabels[Join[
Transpose[{VertexPoints[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Transpose[{Midpoints[t],{"(Subscript[M, 1])","(Subscript[M, 2])","(Subscript[M, 3])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[tp],{"(Subscript[P, 1])","(Subscript[P, 2])","(Subscript[P, 3])"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[tq],{"(Subscript[Q, 1])","(Subscript[Q, 2])","(Subscript[Q, 3])"}}]
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]