Generalizing the property of two parabolas tangent to a circle, each of which touches it at two points

Question

About five years ago, I discovered a property of a parabola, a circle that touches it at two points, and two tangents to it that are parallel to the index of the parabola. In the previous picture $AC⊥BD$

After some trying, I was able to prove this with analytical geometry

But my happiness was soon complete until I was able to generalize this property, as it was not required for two parallel lines to be parallel to the index of the parabola.

I was also able to prove this with analytical geometry

But after a while I noticed that I could consider two parallel lines to be a special case of a parabola when the distance between the focus and the guide approaches zero and they are located at infinity. The property is actually valid for two parabolas

Then, after a while, I found that this property was already known and was present in Arseniy Akuban’s book

This was frustrating, as I had used the previous properties in many applications and theorems

But after a while I was able to generalize this feature more:

I discovered that these two perpendicular lines are nothing but the interior bisector and exterior bisector of the angle between the diagonals of the quadrilateral whose vertices are the four points of contact, as shown in the picture:

Reaching this stage took almost a year of coming up with one generalization after anotherTheorems have been used along the way in many engineering constructions

But I still do not know how to prove this last general case. It is really difficult. I would be grateful for any help in proving this, and I prefer engineering methods to analytical methods.

Is the latest circular known?

Answer 1

Still not an answer, but I would like to share a generic SAGE program (SAGE is a free Python-like online resource ; see below this program and the way to run it). Writing this program has in fact be beneficial because of the need to compute analytical results that could be useful for reaching the assigned goal.

Fig. 1.

The principle is based on the following remark : parabolas externaly tangent to the unit circle having a vertical-axis constitute a family $(F)$ with common cartesian equation :

$$y=kx^2+s(k) \ \ \ \text{with this shift :} \ \ \ s(k):=-k-\frac{1}{4k}$$

depending on a single parameter $k$.

We can assume WLOG that the circle is the unit circle. Let us choose two parabolas $P_1$ and $P'_2$ of family $(F)$ with resp. parameters $k_1$ and $k_2$. $P_1$, featured in red on the figure, will be kept as it is. The second parabola $P_2$, will be the image by a rotation with angle $a$ of $P'_2$ (transformation of the dotted-blue parabola into the solid-blue parabola).

A result among others that I have found is that the point of intersection is

$$\left(\frac{\cos(a)w(k_1)-w(k_2)}{\sin(a)}, w(k_1)\right) \ \text{with} \ w(k)=-\tfrac{1}{2k}$$

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Here is how one can execute this program :

• ask https://sagecell.sagemath.org/

• copy-paste the program in the edit window,

• modify some parameters like $k_2$ (stretching of the blue parabolas) and/or rotation angle $a$.

• execute the program by clicking on the "Evaluate" button.

  var('x')
  k1=2;k2=3/2;a=2*pi/3; # parameters for parabolas P1 and P2
  L=2.5; # min and max values are -L < x,y < +L
  c=cos(a);s=sin(a)
  def w(k) : # ordinate of ref. chord
     return(-1/(2*k))
  def v(k) : # parabolas' vertical shift
     return(-k-1/(4*k))
  def h(k) : # half-chord length
     return(sqrt(1-1/(4*k^2)))
  def p(k,x) :
     return(k*x^2+v(k))
  def R(x,y) : # direct rotation
     return([c*x-s*y,s*x+c*y])
  def S(x,y) : # inverse rotation
     return([c*x+s*y,-s*x+c*y])
  g=circle((0,0),1,color='green') # unit circle
  g+=plot(p(k1,x),(x,-L,L),color='red') #  parabola P1
  u=p(k2,x)
  g+=plot(u,(x,-2,2),linestyle='dotted') # parabola P2 before,...
  g+=parametric_plot(R(x,u),(x,-L,L)) #  then after, rotation.
  g+=line([[-h(k1),w(k1)],[h(k1),w(k1)]],color='red') # chords 
  g+=line([R(-h(k2),w(k2)),R(h(k2),w(k2))])
  # equation giving the abscissas x of points of intersection of P1 and P2 :
  h=S(x,p(k1,x));sol=solve(p(k2,h[0])==h[1],x)
  # ordered list of abscissas of these points :
  X=[sol[n].rhs().real().n(20) for n in range(len(sol))]
  X.sort()
  Y=list(map(lambda x: p(k1,x), X)); # list of ordinates
  # the 2 diagonals with endpoints
  D=[[X[0],Y[0]],[X[2],Y[2]]]
  g+=line(D,color='cyan');g+=point(D,color='green',size=50)
  D=[[X[1],Y[1]],[X[3],Y[3]]]
  g+=line(D,color='cyan');g+=point(D,color='green',size=50)
  # common intersection point
  g+=point([(c*w(k1)-w(k2))/s,w(k1)],color='blue',size=50)
  g.set_axes_range(-L,L,-L,L)
  show(g)
  

Answer 2

As for the properties resulting from special cases and their proof, I have discussed them in Arabic. I will translate them partially later, if he wishes. I will put them in picture form now... Steps to construct the tangent circle to a parabola at two points given one of The middle of the two contact points:

But I could not prove the case of perpendicularity of the two parabolas mentioned in Arseniy Akopyan's book. There are a few other details that I did not include in the pictures, but that is most of the work.

As for constructing a parabola that is tangent to a circle at two points, keep in mind that the center of the circle is a distance from the middle of the two tangent points, a distance equal to the distance between the focus and the index in the parabola. The center of the circle passing through the center of the circle tangent to a parabola at two points and passing through the two points of tangency will be the focus of the parabola.