MorleyTheoremNotebook::usage="From a base triangle, the trisectors of their internal angles are drawn. The adjacent trisectors intersect in pairs and form an equilateral triangle inside. Enter in the command below, the coordinates in clockwise direction. Reference: Weisstein, Eric W. Morley's Theorem , From MathWorld; Weisstein, Eric W. First Morley's Triangle, From MathWorld; Personal";
In[]:=
MorleyTheoremNotebook[{x1_,y1_},{x2_,y2_},{x3_,y3_}]:=​​Module{},Test=Det[{{x1,y1,1},{x2,y2,1},{x3,y3,1}}];​​IfTest==0,Print[PasteButton["It's not a triangle."]],​​P1={x1,y1};P2={x2,y2};P3={x3,y3};​​aa=
2
(x3-x2)
+
2
(y3-y2)
;bb=
2
(x3-x1)
+
2
(y3-y1)
;cc1=
2
(x2-x1)
+
2
(y2-y1)
;​​α=ArcCos
2
bb
+
2
cc1
-
2
aa
2bbcc1
;β=ArcCos
2
aa
+
2
cc1
-
2
bb
2aacc1
;γ=ArcCos
2
aa
+
2
bb
-
2
cc1
2aabb
;​​xx=N[aa(1)];yy=N2bbCos
1
3
γ;zz=N2cc1Cos
1
3
β;​​
λ
1
=
xx
xx+yy+zz
;
λ
2
=
yy
xx+yy+zz
;
λ
3
=
zz
xx+yy+zz
;​​PuntoD=
λ
1
{x1,y1}+
λ
2
{x2,y2}+
λ
3
{x3,y3};​​xd1=Take[PuntoD,{1}].{1};yd1=Take[PuntoD,{2}].{1};​​xx1=N2aaCos
1
3
γ;yy1=N[bb(1)];zz1=N2cc1Cos
1
3
α;​​
λ
4
=
xx1
xx1+yy1+zz1
;
λ
5
=
yy1
xx1+yy1+zz1
;
λ
6
=
zz1
xx1+yy1+zz1
;​​PuntoE=
λ
4
{x1,y1}+
λ
5
{x2,y2}+
λ
6
{x3,y3};​​xe1=Take[PuntoE,{1}].{1};ye1=Take[PuntoE,{2}].{1};​​xx2=N2aaCos
1
3
β;yy2=N2bbCos
1
3
α;zz2=N[cc1(1)];​​
λ
7
=
xx2
xx2+yy2+zz2
;
λ
8
=
yy2
xx2+yy2+zz2
;
λ
9
=
zz2
xx2+yy2+zz2
;​​PuntoF=
λ
7
{x1,y1}+
λ
8
{x2,y2}+
λ
9
{x3,y3};​​xf1=Take[PuntoF,{1}].{1};yf1=Take[PuntoF,{2}].{1};​​Print[" "];​​l=Grid[{{
DE
,
DF
,
FE
},{N[Norm[{xd1,yd1}-{xe1,ye1}]],N[Norm[{xd1,yd1}-{xf1,yf1}]],N[Norm[{xf1,yf1}-{xe1,ye1}]]}},ItemSize15,ItemStyle15,FrameAll];​​Print[l];​​Print[" "];​​label[sym_,Point[pos_]]:={PointSize[Scaled[0.018]],Hue[1,1,1],Point[pos],Text[Style[sym,White,FontSizeScaled[0.012]],pos]};​​Show[{Graphics[{LightYellow,EdgeForm[{Thick,Black}],Polygon[{P1,P2,P3}]}],Graphics[{Green,EdgeForm[{Thick,Black}],Polygon[{{xe1,ye1},{xd1,yd1},{xf1,yf1}}]}],​​Graphics[{Blue,Thick,Line[{P3,{xd1,yd1}}]}],Graphics[{Blue,Thick,Line[{P2,{xd1,yd1}}]}],​​Graphics[{Purple,Thick,Line[{P3,{xe1,ye1}}]}],​​Graphics[{Purple,Thick,Line[{P1,{xe1,ye1}}]}],​​Graphics[{Orange,Thick,Line[{P1,{xf1,yf1}}]}],​​Graphics[{Orange,Thick,Line[{P2,{xf1,yf1}}]}],Graphics[{MapThread[label,{{"A","B","C","D","E","F"},{Point[P1],Point[P2],Point[P3],Point[{xd1,yd1}],Point[{xe1,ye1}],Point[{xf1,yf1}]}}]}]},AxesTrue,BoxRatiosAutomatic,AxesOrigin{0,0},ImageSize900,PlotRangeAll]​​
In[]:=
MorleyTheoremNotebook[{-2,5},{7,12},{9,-4}]
DE
DF
FE
2.43371
2.43371
2.43371
Out[]=