overlapping circles with diameters = tangents?

If we are given:

circles O1 and O2 which meet exactly at the points {A,D} so that the diameter AB of O1 is tangential to O2 at the point A, and the diameter AC of O2 is tangential to O1 at the point A (and so AB is perpendicular to AC),

then

the question is: does D lie on the line BC?

To find out, I used A as the origin of a Cartesian coordinate system, with the diameters lining up with the positive x and y axes respectively.

The equation of the line AC is immediate.

The approach of finding the points of intersection of the two circles (in order to see if it lay on the line) was not feasible.

Another approach was to find the points of intersection of L with each of the circles, and then see if the answers coincided.

If I did not make a mistake, then the answers did not coincide unless the diameters were equal.

This is frustrating, as the gut feeling, bolstered by various drawings of various diameters always appearing to show that D lies on AC.

So, 2 questions:

(a) is this result known?

(b) is there an easier way to do this?

(c) am I wrong, and somehow D does lie on AC?