ProjectionOfTheOrigin::usage="A point A on the X axis moves in the positive direction. The circle has its center at the origin and radius R. A straight line is drawn from point C of tangency to point A. Prove that the projection of the origin on the bisector of the angle OAC as A moves is in the line y =
R
2
.​
​​​References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; Personal."
Development:
Coordinates of point C:
C={x,y};O={0,0};A={a,0};​​

OC
·

AC
==0;
In[]:=
{x,y}.{x-a,y}==0
x(-a+x)+
2
y
0;​​
2
x
+
2
y
-ax==0;​​
2
R
-ax==0;​​x==
2
R
a
;​​y=
2
R
-
4
R
2
a
==
R
a
2
a
-
2
R
;​​C=
2
R
a
,
R
a
2
a
-
2
R
;
Parametric equations of the bisector of angle OAC:
Unit vectors

AC
and

AO
:
In[]:=
AC=
1
2
2
R
-
2
a
a
+
2
R
a
2
a
-
2
R

2
R
a
-a,
R
a
2
a
-
2
R
;​​AO={-1,0};​​AC+AO
In[]:=
-1+
-a+
2
R
a
2
R

2
a
-
2
R

2
a
+
2
-
2
a
+
2
R

2
a
,
R
2
a
-
2
R
a
2
R

2
a
-
2
R

2
a
+
2
-
2
a
+
2
R

2
a
//FullSimplify
-
a+
2
a
-
2
R
a
,
R
a
;
ℒ={a,0}+t-
a+
2
a
-
2
R
a
,
R
a
;
Projection of the origin on the bisector line:
Point P’ is on ℒ:
In[]:=
P'={a,0}+t-
a+
2
a
-
2
R
a
,
R
a
;​​OP'={a,0}+t-
a+
2
a
-
2
R
a
,
R
a
;​​SolveOP'.-
a+
2
a
-
2
R
a
,
R
a
==0,t//FullSimplify
t
a
2
;
In[]:=
{a,0}+
a
2
-
a+
2
a
-
2
R
a
,
R
a
//FullSimplify

1
2
a-
(
2
a
-
2
R
)
,
R
2
;
The component “y” is:
y=
R
2
;
This verifies the statement.
Animation:
For the animation it is considered R =2:
In[]:=
ManipulateO1={0,0};C1=
4
a
,
2
a
2
a
-4
;P1=
1
2
a-
2
a
-4
,1;A={a,0};Line1[x_]:={x,1};Line2[t_]:={a,0}+t-
a+
2
a
-4
a
,
2
a
;​​Show​​ParametricPlotCalloutLine1[x],"y =
R
2
",{Scaled[0.75],Above},LabelStyle{14,Bold,Black},{x,-6,6},PlotStyle{Black,Dashed},​​ParametricPlot[Line2[t],{t,-7,5},PlotStyle->{DarkGreen,Thick}],​​Graphics[{Black,PointSize[0.010],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[P1],Text[Style["P'",14,Bold],P1,{0,-1.2}],​​Point[C1],Text[Style["C",14,Bold],C1,{0,-1.2}],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],Thick,InfiniteLine[{O1,A}],Blue,Circle[O1,2],Red,InfiniteLine[{C1,A}],Dashed,DarkGreen,Line[{O1,P1}]}],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-5,5},{-5,5}},ImageSize700,Style["P' is the origin projected in the bisector line ",Bold,Large],{{a,2.100002,"Value (a)"},2.1,5,0.00001},ControlPlacementTop
Out[]=
​
P' is the origin projected in the bisector line
Value (a)