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Two overlapping circles with tangents drawn at their intersection points intersecting at each others' centres.

So I'm stumped by what should be a rather simple problem. There are two circles whose tangents intersect at each others' centres. The tangents are at right angle. If I know the distance between the centres, there should be simple geometry to solve the radii of the circles.

I know I could do like an equation group of this; call radius one $a$, radius two $b$, write down Pythagoran theorem for the triangle in the middle, then maybe like trigonometric functions for the halves of the two central angles of the circles, but I can't believe it should be this complex (that would be what, a four equation group?). There's something about the symmetry of these circles I'm missing that ought to make this simpler.

I know I have: Let the radius of circle $A$ be $a$, and circle $B$ be $b$. Further, let the line drawn at their intersection be $c$, and the line from $c$ to centre of circle $B$ be $d$. $$a^2 + b^2 = 10^2 \\a^2 + (10-d)^2 = \left( \frac c 2 \right) ^2 \\ b^2 + d^2 = \left( \frac c 2 \right) ^2 $$

If I add to that trigonometric functions and the sum of the angles, I believe I could solve them but it just feels way too complex and certainly not the natural solution.

An aside: does anyone know how these circles (ones whose tangents drawn at the intersection points of the circles are the centres of one another) are called? I know there's a term for it but I can't find it for the life of me.

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  • $\begingroup$ Even if you constrain the centers to be a distance of $10$ apart there's a one-parameter family of consistent solutions since you can hold the distance to $10$ but move $a$. $\endgroup$ – Neal Apr 23 at 14:16
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They are called orthogonal circles.

The necessary and sufficient condition for two circles to be orthogonal can be given under two different forms :

$$R^2+R'^2=d^2 \ \ \ \ \iff \ \ \ \ aa'+bb'=\tfrac12(c+c')\tag{1}$$

where $R,R'$ are their radii,

$$d=\sqrt{(a-a')^2+(b-b')^2} \ \ \text{ the distance between their centers}$$ and

$$\begin{cases}x^2+y^2-2ax-2by+c&=&0\\ x^2+y^2-2a'x-2b'y+c'&=&0\end{cases} \ \ \ \text{their cartesian equations}$$

There is a very nice representation of (1) in the so-called "space of circles", where a circle with equations

$$x^2+y^2-2ax-2by+c=0 \ \ \ \iff \ \ \ (x-a)^2+(y-b)^2=R^2\tag{2}$$

is represented by 3 coordinates, $(a,b,c)$.

Please note the relationship :

$$a^2+b^2-R^2=c\tag{3}$$

If we write (3) under the form :

$$R^2=\underbrace{a^2+b^2-c}_{\|\sigma\|^2}\tag{3}$$

it gives us the opportunity to define the norm of a circle (nothing scandalous : the norm of a circle is plainly its radius).

We now define the dot product between 2 circles with coordinates $(a,b,c)$ and $(a',b',c')$ by :

$$\sigma \ \cdot \ \sigma' \ := \ aa'+bb'-\dfrac12(c+c'). \tag{4}$$

One can show easily the nice relationship:

$$\sigma \ \cdot \ \sigma' \ = \|\sigma\|\|\sigma'\| \cos \alpha$$

(proof below)

with $\alpha$ defined as the angle between the radii at intersection point $I$.

Particular case : is $\alpha =\dfrac{\pi}{2}$, we find relationship (1) !

enter image description here

Fig. 1 : 2 non-orthogonal circles illustrating the notations for relationship (4).

Appendix : Proof of relationship (4).

The law of cosines in triangle $OIO'$ gives:

$$d^2=R^2+R'^2-2RR' \cos \alpha$$

which can be written :

$$(a-a')^2+(b-b')^2=a^2+b^2-c+a'^2+b^2-c-2RR' \cos \alpha$$

Expanding the LHS and simplifying, we get indeed :

$$2aa'+2bb'-(c+c')=2RR' \cos \alpha$$

which nothing else than (4).

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  • $\begingroup$ Thanks for the detailed answer! So in short, there's insufficient information to solve the problem, and we can't infer the ratio of R and R' from just the fact that the circles are orthogonal. $\endgroup$ – Jarno Porkka Apr 24 at 14:27
  • $\begingroup$ Indeed, the ratio R'/R can take any value. See for example the case of Appolonian circles given in this answer of mine $\endgroup$ – Jean Marie Apr 24 at 16:16
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You can't solve for the individual circle radii without more information – you only have $a^2+b^2=10^2$ to go by. There is one degree of freedom in the problem.

The two circles in the diagram may be called orthogonal, as they intersect at right angles.

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Let the centers of the circles be $O_1$ and $O_2$. Draw a circle with the diameter $O_1O_2$. Take an arbitrary point $C$ on the circle. Draw through $C$ two circles centered at $O_1$ and $O_2$. The circles will be orthogonal (i.e. correspond to the requirement that the tangent of one circle drawn at intersection point goes through the center of the other circle). This construction proves that the given data are not sufficient to determine unambiguously the radii of the circles.

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  • $\begingroup$ Thanks. I suppose there's something wrong with the whole setup here. I'll have to go back to the drawing board. $\endgroup$ – Jarno Porkka Apr 24 at 14:17

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