Toothpick Construction of a Square: What is it?
A Mathematical Droodle
What if applet does not run? 
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Copyright © 19962018 Alexander BogomolnyToothpick Construction of a Square
Matchsticks and a rusty compass serve formidable geometric tools to play with. The applet below suggests a construction of a right angle and a square using only toothpicks. (Matchsticks would of course perform the same job as well.) The rules of the toothpick constructions allow placing toothpicks endtoend or at certain marked points. The best the endtoend construction can do is to produce an equilateral triangle or, provided there is a sufficient supply of the material, a triangular lattice, or grid, whose algebraic properties have been discussed elsewhere. To enable nontrivial constructions we mark a point somewhere on a stick and allow placing toothpicks at this point.
What if applet does not run? 
In the applet, AB is the starting toothpick, with point C marked on it. All other points are named alphabetically in the order of their construction.
Points D and E are obtained by two toothpicks joint at their endpoints with the other ends at points A and C. Let's denote equal angles CAD and CAE as α. By constructing successively three equalateral triangles AEF, AFG, and AGH, we get a straight line segment EAH. We add another equilateral triangle ADI and note the point J of intersection of toothpicks GH and DI. We show that angle BAI is right, independent of the position of point C on AB.
Indeed,

Since, also ∠BAD = α, the two toothpicks, AH and AD are symmetric in a line perpendicular to AB at A. (Both angles HAJ and DAJ equal 90°  α.) Thus we may place a toothpick AK over point J. ∠KAB is right. When two toothpicks, one with an end at B and the other at K, have their other ends touch, they touch at L such that BAKL is a square.
We got a funny construction of a square. But it must be taken seriously. Chapter 8 in [Martin] ends with a theorem that states that any point that can be constructed with a ruler and compass, can be constructed with a supply of toothpicks, provided a point is marked on one of them.
References
 G. E. Martin, Geometric Constructions, Springer, 1998, Chapter 8.
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