Uninverted Diagram:

Inverted Diagram (about point ):


Looking at point in our uninverted diagram, we have almost immediate motivation to invert about this point, since doing so will bring our messy tangent circles to lines. We can do so, and this is illustrated in the second diagram.

We claim that this inversion will send circles and to parallel lines. We know and will become lines perpendicular to the line between the centers of those circles and ... but this just means the lines are perpendicular to the same line, so they must be parallel. Similarly, we conclude that and must also invert to a pair of parallel lines. Thus, the inversions of our four tangent circles about go to a parallelogram.

We haven't yet shown where is yet, but we know where , , , will go. Noting that line does not pass through and contains point , we see that inverts to a circle through . Hence are concyclic, and similarly, are concyclic. Thus is the intersection of the circumcircles of and .

The angle condition hasn't been dealt with yet. Let's see what this is under inversion. We can decompose this condition into:

Under inversion, we can easily see that:

Also, in our inverted diagram, since we have a parallelogram, we can easily claim that .

With this information, we claim are collinear. The proof is a simple angle chase, denoting as the angle going from to , and as the angle going the other direction:



Hence we can easily see that , which proves our claim.

Noting this, we can show that are also collinear. This is another angle chase, splitting .
Then, with help from a couple of cyclic quads, and our parallel lines:




Hence is a line, and we can note that is a circle through , , , and . Therefore, since a line not passing through the center of inversion goes to a circle through it, we know that are concyclic, completing the problem. .