Tuesday, May 19, 2020

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 + r2 = r3), 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.


No comments:

Post a Comment