# 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:

• First draw a circle, about each point, that passes through the other.
• Now connect the two points by one line and the two points where the circles met by another; this is the perpendicualr bisector of the first; I'll call the point where they meet C. I'll use, as unit, the length of each half into which this cuts the first – the shared radius of the two circles – so the circles have radius two and the perpendicular bisector of the shared radius has length 2.√3, which the shared radius bisects.
• Draw a circle, centred on one of the points where the first two circles meet, that passes through both the initial points. (This cuts the original circles in points opposite one another; and the 2.√3 line is the perpendicular bisector of the diameter between these points.)
• Connect one of the initial points, A, through the centre of this third circle to its far side, to form a diameter; connect the far end of this to the other of the original two points, B; the straight line that does this is perpendicular to the original straight line from A to B. The part of this line that's a radius of the circle about B has length two units and is perpendicular to the line CB, which has length one unit; so the line from C (the mid-point of AB) to where this newest line cuts the circle about B has length √5.
• Draw a circle about C that passes through that meeting point, so has radius √5. Extend AB to meet this circle on B's side and the original circle about A on the other side; the extension cuts the newest circle at two points; one is (√5)−1 units from B on the side opposite A; the other is (√5)+1 units from B on the other side of A from B.
• Draw a circle about B through the more distant of those two points, on the far side of A; this circle has radius 1+√5; the points where it cuts the original circle about A are vertices of a pentagon, along with the point opposite B on that circle.
• Extend the AB line to the second point where it meets our 1+√5 circle about B, on the side of B opposite A; connect this point by straight lines to the two points where the 1+√5 circle cut the original circle about A. The points where these lines meet the A-circle along the way are the other two vertices of our pentagon; indeed, the parts of these two lines that lie inside the A-circle are two of the edges of the pentagon.
• Connect these last two points to each other and the first two vertices of the pentagon to the point opposite B on the A-circle; you have now completed a pentagon inscribed in the circle we originally drew about A.
• This takes five circles (three of radius 2, one each of radius √5 and 1+√5), four construction lines (the shared radius, extended, its perpendicular bisector, a diameter of the third circle and the line from its point opposite A to B; this last being perpendicular to AB) and the five edges of the pentagon (two of them extended).

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.

Written by Eddy.