TriangleConstantArea::usage=" Given two lines that intersect at the origin and in the plane of these two lines, a third PQ line of variable length moves so that the triangle POQ formed by the three lines has a constant area. At point P where the line PQ cuts to the top line, a perpendicular is drawn and at point Q where the line PQ cuts to the bottom line, a perpendicular is drawn. Obtain the geometric locus of the point M of intersection of the perpendiculars. References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; Personal.​
";
Development:
Be the straigh lines: y =
1
2
x and y =
-
1
2
x . The relationship between the x coordinates must be inversely proportional for the area to be constant. In our example the area will be 2. When a vertex is the origin, the area of the triangle is calculated as follows:
A=
1
2
(
x
1
y
2
-
x
2
y
1
)
. To obtain a constant area, the following must be met: 
x
1
x
2
=2A.
In[]:=
​​P={4x1,2x1};Q=
1
x1
,-
1
2x1
;​​A=
1
2
Abs4x1-
1
2x1
-
1
x1
(2x1)
In[]:=
2;
Geometric locus of M:
Perpendicular to the OP line in P:
In[]:=
y-2x1==-2(x-4x1);
Perpendicular to the OQ line in Q:
In[]:=
y+
1
2x1
==2x-
1
x1
;
In[]:=
Solvey-2x1==-2(x-4x1),y+
1
2x1
==2x-
1
x1
,{x,y}
x
5(1+4
2
x1
)
8x1
,y
5(-1+4
2
x1
)
4x1
;​​M==
5(1+4
2
x1
)
8x1
,
5(-1+4
2
x1
)
4x1
;
Animation:
In[]:=
ManipulateO1={0,0};M[t_]:=
5(1+4
2
t
)
8t
,
5(-1+4
2
t
)
4t
;P={4x1,2x1};Q=
1
x1
,-
1
2x1
;Line1[x_]:=x,
1
2
x;Line2[x_]:=x,-
1
2
x;A=
1
2
Det
1
x1
,-
1
2x1
,{4x1,2x1};​​w=Grid[{{Area},{A}},FrameAll,ItemStyle->15,ItemSize15];​​Show[{​​ParametricPlot[M[t],{t,0.5,x1},ColorFunction->"BlueGreenYellow"],​​ParametricPlot[Line1[x],{x,-21,21},PlotStyle{Green,Thick}],​​ParametricPlot[Line2[x],{x,-21,21},PlotStyle{Red,Thick}],​​Graphics[{Black,PointSize[0.010],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[P],Text[Style["P",14,Bold],P,{0,-1.2}],​​Point[Q],Text[Style["Q",14,Bold],Q,{0,-1.2}],​​Point[M[x1]],Text[Style["M",14,Bold],M[x1],{0,-1.2}],​​Thick,Line[{Q,P}],Dashed,Green,Line[{P,M[x1]}],Red,Line[{Q,M[x1]}]}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-1,8.5},{-1,10}},ImageSize700],Style["Geometric locus of M ",Bold,Large],{{x1,0.5+0.00002,"Value (​
x
1
​)"},0.5+0.00001,2,0.00001},Delimiter,{{w,3,"Area of triangle POQ"}},ControlPlacementTop
Out[]=
​
Geometric locus of M
Value (​
x
1
​)
Area of triangle POQ
Area
2.
The geometric locus of point M is of hyperbolic type.