MpointHypotenuse::usage="Let C be a circle with a center in O and radius R, and a P a point inside C. Two perpendicular lines are drawn PA and PB that pass through P, where A and B are the points of intersection of said lines with the circle C. Find the geometric locus of the M that divides the AB side into a given rate r, when it turns.​
​References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; Personal."
Development:
Calculations for point B:
P={0,0};O={a,0};A={RCos[θ]+a,RSin[θ]};​​Solvey==-
RCos[θ]+a
RSin[θ]
x,
2
(x-a)
+
2
y
==
2
R
,{x,y}//FullSimplify
x
1
2(
2
a
+
2
R
+2aRCos[θ])
2a
2
R
-
2
2
R
(-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ]))
2
Csc[θ]

2
Sin[θ]
,y
(a+RCos[θ])-a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
Sin[θ]
R(
2
a
+
2
R
+2aRCos[θ])
,x
a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
2
Sin[θ]
2
a
+
2
R
+2aRCos[θ]
,y-
(a+RCos[θ])a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
Sin[θ]
R(
2
a
+
2
R
+2aRCos[θ])
;
Animation:
The coordinates of the B and M points are in the animation.
For the animation it is considered R = 3 and a = 1:
In[]:=
Manipulate​​R=3;a=1;​​P={0,0};O1={1,0};​​​​xA[θ_]:=N[RCos[θ]+a];​​​​yA[θ_]:=N[RSin[θ]];​​A[θ_]:={xA[θ],yA[θ]};​​​​xB[θ_]:=N
a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
2
Sin[θ]
2
a
+
2
R
+2aRCos[θ]
;​​yB[θ_]:=N-
(a+RCos[θ])a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
Sin[θ]
R(
2
a
+
2
R
+2aRCos[θ])
;​​xB1[θ_]:=N
1
2(
2
a
+
2
R
+2aRCos[θ])
2a
2
R
-
2
2
R
(-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ]))
2
Csc[θ]

2
Sin[θ]
;​​yB1[θ_]:=N
(a+RCos[θ])-a
2
R
+
2
R
-2
4
a
+
2
a
2
R
+2
4
R
-aR(4(a-R)(a+R)Cos[θ]+aRCos[2θ])
2
Csc[θ]
2
Sin[θ]
R(
2
a
+
2
R
+2aRCos[θ])
;​​B[θ_]:={xB[θ],yB[θ]};​​B1[θ_]:={xB1[θ],yB1[θ]};​​​​M[θ_]:=
xA[θ]+rxB[θ]
1+r
,
yA[θ]+ryB[θ]
1+r
;​​M1[θ_]:=
xA[θ]+rxB1[θ]
1+r
,
yA[θ]+ryB1[θ]
1+r
;​​r=
1
n
;​​Show[{​​ParametricPlot[M[θ],{θ,0,θ1},ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality"],​​ParametricPlot[M1[θ],{θ,0,θ1},ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality"],​​Graphics[{Black,PointSize[0.010],​​Point[P],Text[Style["P",14,Bold],P,{0,-1.2}],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[A[θ1]],Text[Style["A",14,Bold],A[θ1],{0,-1.2}],​​Point[B[θ1]],Text[Style["B",14,Bold],B[θ1],{0,-1.2}],​​Point[B1[θ1]],Text[Style["
B
1
",14,Bold],B1[θ1],{0,-1.2}],​​Point[M[θ1]],Text[Style["M",14,Bold],M[θ1],{0,-1.2}],​​Point[M1[θ1]],Text[Style["
M
1
",14,Bold],M1[θ1],{0,-1.2}],​​Thick,{Line[{A[θ1],B[θ1]}],Dashed,Line[{A[θ1],B1[θ1]}]},Blue,Circle[O1,R],Red,InfiniteLine[{P,B[θ1]}],InfiniteLine[{P,A[θ1]}]}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-5,5},{-5,5}},ImageSize700],Style["Geometric locus of M",Bold,Large],{{θ1,0.00002,"Value (θ)"},0.00001,2π,0.00001},Delimiter,{{n,1,"Value (n)"},1,10,1},ControlPlacementTop
Out[]=
​
Geometric locus of M
Value (θ)
Value (n)