## Proposition 46

We now have the second regular polygon, the first being the equilateral triangle of proposition I.1. Regular polygons with 5, 6, and 15 sides are constructed in Book IV.

#### Consruction steps

 There are quite a few steps needed to construct a square on AB. In order to construct the perpendicular AC, first AB has to be extended in the direction of A and a point F on the far side the same distance from A as B is, then two more circles centered at B and F to get a perpendicular line, and then it needs to be cut off at length C, but fortunately, the needed circle has already been drawn.
Next, EB is to be drawn through B parallel to AD. In general that construction given in I.31 takes six circles, but in this case if EB is drawn perpendicular to AB, then it will be parallel to AD, too, and that construction only takes three circles with radii BA, AG, and GA.

This abbreviation of Euclid's construction requires six circles and four lines. There are alternate constructions that are a bit shorter. For instance, E may be found as the other intersection of the circles of radii BA and DA.

#### Use of Proposition 46

The construction of a square given in this proposition is used in the next proposition, numerous propositions in Book II, and others in Books VI, XII, and XIII.

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