Puzzle: Simple Construction?
Last week I taught a class to construct a tangent to a circle from a point outside the circle (center O) (construct the segment joining the segment and the center, find the midpoint of that segment (M), construct a circle with center M and radius MO. The new circle intersects the original circle twice, either one of those points taken with the external point is a tangent.)
Can you construct the external tangent to two circles?
So, they get the explanation, more or less (we’ve effectively inscribed an angle in a semicircle) and practiced the construction, and I wonder out loud, why don’t we construct the external tangent to two circles? And I throw them on the board, and someone says to connect the centers, and I agree, and pause.
“Not today guys, I think I know where to go, but I am not sure. We’ll look at this one next week.”
So I call my geometry expert, McRib. “Easy!” he says, and starts to tell me, and then pauses. “I can do it with ratios, but I need to think about a nicer construction. Let me get back to you.”
I ended up looking it up. And McRib called back a day and a half later, with a construction.
But without looking it up, can you construct the external tangent to two circles?
Did you figure it out, or did you know it?
JD2718 
I figured out one method a couple years ago when I was teaching this same topic at a summer math camp. My solution was to reduce it to the previously solved problem of constructing the tangents from a point to a circle. I’m curious to learn if there are more direct ways!
I know it, and I posted the method here: http://proooof.blogspot.com/2007/12/tangenti-comuni-due-circonferenze.html
(I forgot to mention that there are, in general, 4 tangents to two circles)
I see the ratios and similar triangles. JD, you can’t do this. I got so much to do, and guess what I’m doing instead!
I know I’ve seen it done before, but I’m pretty sure the method that dropped out from a quick sketch is not the way I previously saw it done… I went from the similar triangles and used the circle theorem on intersecting chords:
http://www.rfbooth.com/dualtangent.html
Looks nice.
I still prefer my approach (borrowed from someone else). Given circle with center O and radius R and another circle with center P and radius r, wlog R > r, construct circle with radius (R – r) and center O. Next construct tangent from P to the new circle (call the pt of tangency T).
Extent OT to the larger circle at T’. T’ is one of our two points of external tangency. Find T’P and construct congruent segment OT” (T” lies on the original smaller circle). T’T” is an external tangent.
That’s beautiful; I’ve added it to my little collection of methods on the page I linked before :).