Constructing the pentagon

I found a web-site with fun geometry challenges using only circle and straight line construction. It took me a little while to solve all their puzzles. I am particularly pleased with my solution to their pentagon problem.

Here's the sequence to build the pentagon in 14 lines:

That's 14 moves, beating the tighter target of 15 the site sets; I also beat the tighter targets on the dodecagon, octagon, hexagon and circle pack 7 puzzles, equalling their tighter targets on all others (albeit some of these weren't solutions inside the origin circle, as above for the pentagram; and getting circle pack 7 in its origin circle in the requested 14 required cunning). For both the octagon and the dodecahedron, it helps to realise that the edges of the final figure, when extended, meet each other on various circles around the figure; once you have two edges that meet on the largest of these, you can draw this circle; lots of edges of the final figure then drop out just by connecting points on the circle to each other or to vertices of the figure; and, since these lines are just extended edges of the figure, they cost nothing, making the circle a very useful construction line. Using this technique, building on the pentagon construction above, it's possible to make a decagon in 22 steps.

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