In[]:=
GeometricLocusVertex::usage="Find the geometric locus of the vertex of the right angle of a right triangle if the ends of the hypotenuse slide over the coordinates axes (see figure). References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; Personal.​
";
Development:
Geometric locus equation:
In[]:=
A={a,0};B={0,b};M={x,y};​​Expand[
2
Norm[M-A]
+
2
Norm[M-B]
==
2
Norm[A-B]
]
Out[]=
2
Abs[-1+x]
+
2
Abs[x]
+
2
Abs[y]
+
2
Abs[-b+y]
1+
2
Abs[b]
In[]:=
2
Abs[x]
+
2
Abs[-a+x]
+
2
Abs[y]
+
2
Abs[-b+y]

2
Abs[a]
+
2
Abs[b]
;​​
2
(x-a)
+
2
(y-b)
+
2
x
+
2
y
-(
2
a
+
2
b
)==0;​​
2
x
+
2
y
-ax-by==0;​​
2
x-
a
2
+
2
y-
b
2
==
2
a
+
2
b
4
;​​ζ[t_]:=
2
a
+
2
b
2
Cos[t]+
a
2
,
2
a
+
2
b
2
Sin[t]+
b
2
;
It is a family of circles.
In[]:=
Manipulateζ[t_]:=
2
a
+
2
b
2
Cos[t]+
a
2
,
2
a
+
2
b
2
Sin[t]+
b
2
;​​A={a,0};B={0,b};tri={A,B,ζ[t]};​​Show[{​​ParametricPlot[ζ[t1],{t1,0,2π},ColorFunction->"BlueGreenYellow"],​​Graphics[{Black,PointSize[0.018],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],​​Point[B],Text[Style["B",14,Bold],B,{0,-1.2}],​​Point[ζ[t]],Text[Style["M",14,Bold],ζ[t],{0,-1.2}],Blue,​​Thick,TriangleConstruct[tri,"Boundary"]}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange12],Style["Family of circumferences",Bold,Large],{{a,-10+0.00002,"Value (a)"},-10+0.000001,10,0.00001},{{b,-10+0.00002,"Value (b)"},-10+0.000001,10,0.00001},{{t,0.00002,"Value (t)"},0.000001,2π,0.00001},AutorunSequencing->All,ControlPlacementTop,SaveDefinitions->True
Out[]=
​
Family of circumferences
Value (a)
Value (b)
Value (t)