Someone might object that we began with the line through A and B. What if we do not want to begin with this line? So our goal is to find a point F such that the distance from A to F is one third that of A to B without using any line.
Mascheroni (1797) showed that all Euclidean constructions could be done just using a compass without a ruler. A few decades later, a work by Georg Mohr in 1672 was discovered proving the same result. (details in Eves: see references)
Here is a construction that uses no lines, just a compass to draw seven circles. (Seven circles is in fact the least number possible: details. Here is a proof that this construction works.)
Beautiful!
I personally love circles, but...
What if we like straight lines and really do not like circles? (going round makes us dizzy, we like to investigate all the angles, we were taught to line up in school!)
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