Mathematics IV: circles

[angus]

Assuming by "touch" you mean "tangent", then one

Clearly there's one in the triangle.  I considered that there might be another circumscribing the whole, but I'm having trouble visualizing how that'd happen given how far the line segment sticks out on the sides.  I'm also assuming a Cartesian plane in Euclid space, etc.  If one draws a circle just the right size, it touches the line segment on the right, and the small circle at about 2 o'clock, and the big circle at about 11 o'clock.  Something like this:



That makes two circles, if it is correct.

If you now place two filled circles inside the two unfilled circles, you get this:



If you then draw a few line segments below the original line segment, attaching them at the ends to make vertices and fill in the resulting polygon, you will get this:



Now, if you add a couple of rectangles, a sliced oval, some color, some more line segments, a hypocycloid of four cusps, and some text, you will end up with this remarkable geometric design.



Fascinating.

[angus]

Yes, I freely admit that I voted for 1, but I am now convinced that this is incorrect.  I am fairly convinced now that there are at least 2.  Unlike all angus polls however, no excelsus poll allows the user to change the vote.  That is your philosophy and I accept that.  Still, I am increasingly convinced, though I did not draw it well, that at least two circles can be drawn which intersect the three loci at exactly one point.  One in the little triangle (the one everyone immediately notices, presumably), and one about the line segment and the two circles, touching the line segment at its right terminus and the two circles roughly where I indicated in the drawing.

Edit:  I see by the time I post this that you have changed the description of your figure.  If the red text is the case, then my original vote was correct.  More importantly, we can now conclude that two important differences exist between angus polls and excelsus polls.  I can accept not allowing users to change vote, but you changing your question in such a poll after we have voted is highly irregular.  

If you want to set your reputation on this forum as a one-trick pony, or as a niche-poster, that's fine with me.  You can be our next waltermitty, although perhaps a slightly better-informed version, and without the fetish for really old women.  But if you are going to ask serious questions using misleading notation, then that's another issue entirely.  You clearly drew a line segment.  You then, after folks voted--in a poll in which you did not allow users to change their votes!--made it clear that you actually intended to make it a line instead of a line segment.  By this we are left to conclude that you are either (a) too ignorant to use the accepted and customary symbols for denoting lines, rays, and line segments, or (b) you are a troll and a trickster.

I'm inclined to believe that both apply.