Two circle problems

A year or so ago I got shown a geometry problem involving 3 circles. Easy to solve. Yesterday, I saw a similar version of it, with just a single change. It was still easy to solve.

Unfortunately, in both cases, both chat-based AIs (Gemini and ChatGPT) failed to solve it. Despite a year or so of progress, the fact that the problem needed a specific first step still confused the models.

What are the problems though? Consider 3 circles of equal radius that are touching such that their centers are colinear. We draw a tangent to the last circle starting from a point on the first one and we need to determine the length of the chord that is determined by the intersection between this line and the middle circle.

Many words, but pictures make this clearer. The first problem is:

And the second one:

In both cases, the solution hinges on determining the distance of the chord in the second circle from the center of that circle. We can build similar triangles:

Once we know \(OD\) in terms of the radius \(R\) of the circles (\(\frac{3R}{5}\) for the second problem and \(\frac{R}{2}\) for the first one), we can just focus on the middle circle:

We know that \(OA\) and \(OB\) are \(R\) and we have two congruent right triangles. Thus \(AB\) is twice \(AD\) which is \(\sqrt{R^2 - OD^2}\). So:

• first problem: \(AB = 2\sqrt{R^2 - R^2/4} = R\sqrt{3}\)
• second problem: \(AB = 2\sqrt{R^2 - 9R^2/25} = 1.6R\)

That’s all. One single crucial step at the first, which the AIs fail to consider.