Circles of Apollonius of a Triangle

Let’s investigate all triangles in which one side length is twice another side length and the third side length is a constant length of 9. Some triangles that fit these requirements would include ones with integer side lengths (4, 8, 9), (5, 10, 9), (6, 12, 9), (7, 14, 9), and (8, 16, 9):

Placing each of these triangles one on top of the other, it would appear that the third vertex follows a circular path:

and can be animated as follows:

Is this path always a circle, or is this just a coincidence with the values that we picked? 

This is no coincidence.  The Greek mathematician Apollonius showed that any circle can be defined as the set of points with a constant ratio of distances to two set points.  Conversely, we can prove that a set of points which maintains a constant non-one ratio of distances to two fixed points will always form a circle. 

If the constant ratio is k = ^b/_a and the distance between the two fixed points is c so that a < b < c, then we can make △ABC with side lengths a, b, and c and with opposite angles ∠A, ∠B, and ∠C, so that ∠B is at the origin and side c is along the x-axis so that the coordinate of A is (c, 0).

Then the locus of points C that are ak away from B and bk away from A can be expressed by the distance equation as x² + y² = a²k² and (x – c)² + y² = b²k².  Rearranging and combining gives a²b²k² = b²x² + b²y² = a²(x – c)² + a²y², and further rearranging gives (x + ^{a^2 c}/_{b^2 – a^2})² + y² = (^abc/_{b^2 – a^2})², a circle equation with a radius of r = ^abc/_{b^2 – a^2}. Therefore, vertex C will always lie on the path of a circle.

In fact, three circles of Apollonius can be drawn for any scalene triangle, one for each vertex:

Using the same argument as above, the radii of the three circles of Apollonius will be r₁ = ^abc/_{c^2 – a^2}, r₂ = ^abc/_{c^2 – b^2}, and r₃ = ^abc/_{b^2 – a^2}.  

Since ¹/_r1 = ¹/_abc(c² – a²), ¹/_r2 = ¹/_abc(c² – b²), and ¹/_r3 = ¹/_abc(b² – a²), and since ¹/_abc(c² – a²) = ¹/_abc(b² – a²) + ¹/_abc(c² – b²), we can derive the elegantly compact relation:

where r₁ < r₂ and r₁ < r₃.

Another way to write this equation is r₂^-1 + r₃^-1 = r₁^-1, which places it in an increasing list of geometric equations that are in the form of xⁿ + yⁿ = zⁿ, the most famous being a² + b² = c², the Pythagorean Theorem.  Other equations in that form include the relationship between the major axis, minor axis, and focal length of ellipses and hyperbolas (b² + c² = a² and a² + b² = c²), the radii of three mutually tangent circles and a line  (r₁^-½ + r₂^-½ = r₃^-½), and, of course, the addition of segments (a¹ + b¹ = c¹).

It is amazing that over a thousand years ago Apollonius showed that any circle can be defined as the set of points with a constant ratio of distances to two set points.  It is also amazing that the simple relationship of ¹/_r1 = ¹/_r2 + ¹/_r3 is true for the radii of the three circles of Apollonius for any scalene triangle, despite the complicated steps in between.