In[]:=
PointOfBrianchon::usage="The point of concurrence of the lines that join the vertices of a triangle and the points of contact of an inconic is known as the Brianchon point. In the case of the MacBeath inconic is the isotomic conjugate of the circumcenter (​
X
264
​). References: Wolfram MathWorld: Brianchon Point, MacBeath Triangle; Clark Kimberling's Encyclopedia of Triangle Centers-ETC; Personal. ";​​​​PointOfBrianchon[tri={{x1_,y1_},{x2_,y2_},{x3_,y3_}},{X0_,X1_},{Y0_,Y1_}]:=​​Module{},DEP=Det[{{x1,y1,1},{x2,y2,1},{x3,y3,1}}];​​P1={x1,y1};P2={x2,y2};P3={x3,y3};IfDEP==0||X0≥X1||Y0≥Y1,Print[PasteButton["Incorrect data or the points do not form a triangle."]],​​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
;​​xx1=N[aaCsc[α]Csc[2α]];yy1=N[bbCsc[β]Csc[2β]];zz1=N[cc1Csc[γ]Csc[2γ]];​​
λ
1
=
xx1
xx1+yy1+zz1
;
λ
2
=
yy1
xx1+yy1+zz1
;
λ
3
=
zz1
xx1+yy1+zz1
;​​BPoint=
λ
1
{x1,y1}+
λ
2
{x2,y2}+
λ
3
{x3,y3};​​xBP=Take[BPoint,{1}].{1};yBP=Take[BPoint,{2}].{1};​​BPoint={xBP,yBP};​​xx2=0;yy2=bb
2
cc1
Cos[γ]Sec[β];zz2=cc1
2
bb
;​​
λ
4
=
xx2
xx2+yy2+zz2
;
λ
5
=
yy2
xx2+yy2+zz2
;
λ
6
=
zz2
xx2+yy2+zz2
;​​ContactPoint1=
λ
4
{x1,y1}+
λ
5
{x2,y2}+
λ
6
{x3,y3};​​xcp1=Take[ContactPoint1,{1}].{1};ycp1=Take[ContactPoint1,{2}].{1};CPoint1={xcp1,ycp1};​​xx3=aa
2
cc1
Cos[γ]Sec[α];yy3=0;zz3=cc1
2
aa
;​​
λ
7
=
xx3
xx3+yy3+zz3
;
λ
8
=
yy3
xx3+yy3+zz3
;
λ
9
=
zz3
xx3+yy3+zz3
;​​ContactPoint2=
λ
7
{x1,y1}+
λ
8
{x2,y2}+
λ
9
{x3,y3};​​xcp2=Take[ContactPoint2,{1}].{1};ycp2=Take[ContactPoint2,{2}].{1};​​CPoint2={xcp2,ycp2};​​xx4=aa
2
bb
Cos[β]Sec[α];yy4=bb
2
aa
;zz4=0;​​
λ
10
=
xx4
xx4+yy4+zz4
;
λ
11
=
yy4
xx4+yy4+zz4
;
λ
12
=
zz4
xx4+yy4+zz4
;​​ContactPoint3=
λ
10
{x1,y1}+
λ
11
{x2,y2}+
λ
12
{x3,y3};​​xcp3=Take[ContactPoint3,{1}].{1};ycp3=Take[ContactPoint3,{2}].{1};​​CPoint3={xcp3,ycp3};​​Line1=InfiniteLine[{P1,P2}];​​Line2=InfiniteLine[{P1,P3}];​​Line3=InfiniteLine[{P2,P3}];​​ContactLine1=InfiniteLine[{CPoint1,P1}];​​ContactLine2=InfiniteLine[{CPoint2,P2}];​​ContactLine3=InfiniteLine[{CPoint3,P3}];​​μ=TriangleCenter[tri,"NinePointCenter"];​​xn=μ[[1]];yn=μ[[2]];​​x4=N[2xn-xcp1];y4=N[2yn-ycp1];​​x5=N[2xn-xcp2];y5=N[2yn-ycp2];​​p5={CPoint1,CPoint2,CPoint3,{x4,y4},{x5,y5}};​​ξ=ResourceFunction["FivePointConic"][p5,{x,y}];​​Print[PasteButton["Information about of the MacBeath inconic and of all points and parameters:"]];Print[" "];​​δ=ResourceFunction["ConicProperties"][ξ==0,{x,y}];​​Print[δ];​​Print[" "];​​p=Grid[{{BrianchonPoint(
X
264
)},{BPoint}},FrameAll,ItemSize->15,ItemStyle->15];​​Print[p];​​Print[" "];​​Show[{ContourPlot[ξ==0,{x,X0,X1},{y,Y0,Y1}],Graphics[{Blue,TriangleConstruct[tri,"Boundary"],PointSize[0.02],Point[P1],Text[Style["
P
1
",14,Bold],P1,{0,-1.2}],Point[P2],Text[Style["
P
2
",14,Bold],P2,{0,-1.2}],Point[P3],Text[Style["
P
3
",14,Bold],P3,{0,-1.2}],Dashed,Line1,Line2,Line3,Black,Point[CPoint1],Text[Style["CP1",14,Bold],CPoint1,{0,-1.2}],Point[CPoint2],Text[Style["CP2",14,Bold],CPoint2,{0,-1.2}],Point[CPoint3],Text[Style["CP3",14,Bold],CPoint3,{0,-1.2}],Dashed,Thick,ContactLine1,ContactLine2,ContactLine3,Red,Point[BPoint],Text[Style["BP",14,Bold],BPoint,{0,-1.2}]}]},AxesTrue,AxesLabel{x,y},AspectRatioAutomatic,AxesOrigin{0,0},PlotRange{{X0,X1},{Y0,Y1}},PlotLabelStyle[Framed["Brianchon point of MacBeath Inconic"],16,Blue,BackgroundYellow]]​​
Example 1:
In[]:=
PointOfBrianchon[tri={{-2,5},{7,12},{9,-4}},{-10,10},{-10,14}]
Information about of the MacBeath inconic and of all points and parameters:
TypeEllipse,Equation34.1398-9.29443x+1.
2
x
-13.8899y+0.318957xy+1.37002
2
y
0,Foci{{6.17722,3.77215},{1.64557,5.4557}},Vertices{{7.78702,3.1741},{0.0357625,6.05375}},Covertices{{5.07952,7.75821},{2.74326,1.46963}},Eccentricity0.584633,Center{3.91139,4.61392},SemimajorAxisLength4.13444,SemiminorAxisLength3.35426,AreaEnclosed43.5676,SemiLatusRectum2.72131,Directrices{y1.0448+2.69173x,y-20.8725+2.69173x},AxesOfSymmetry{y6.06704-0.371508x,y-5.91449+2.69173x},Circumference23.5903,RotationAngle-0.355706,BaseConicEquation0.0585014
2
x
+0.0888803
2
y
1,RotationAngle-0.355706,DisplacementFromOrigin{3.91139,4.61392}
BrianchonPoint
X
264
{2.77899,4.76487}
Out[]=
Example 2:
In[]:=
PointOfBrianchon[tri={{-3,-1},{1,4},{4,-4}},{-5,5},{-5,5}]
Information about of the MacBeath inconic and of all points and parameters:
TypeEllipse,Equation-4.69366-0.585415x+1.
2
x
+0.747914y+0.0716734xy+1.31539
2
y
0,Foci{{1.39362,-0.414894},{-0.787234,-0.170213}},Vertices{{2.50558,-0.53965},{-1.89919,-0.0454564}},Covertices{{0.517877,1.62094},{0.0885065,-2.20605}},Eccentricity0.495111,Center{0.303191,-0.292553},SemimajorAxisLength2.2162,SemiminorAxisLength1.9255,AreaEnclosed13.4061,SemiLatusRectum1.67293,Directrices{y36.7829+8.91304x,y-42.9393+8.91304x},AxesOfSymmetry{y-0.258537-0.112195x,y-2.99491+8.91304x},Circumference13.0276,RotationAngle-0.111728,BaseConicEquation0.203602
2
x
+0.269719
2
y
1,RotationAngle-0.111728,DisplacementFromOrigin{0.303191,-0.292553}
BrianchonPoint
X
264
{-0.170669,-0.343073}
Out[]=
Example 3:
Example 4:
Example 5:
Example 6: