Rulers and Compasses
Perpendicular lines
A circle is drawn from a point
located on line 
, and the intersections between the circle and the line are 
. Circles centered on 
are drawn, and their intersections are 
. The line that connects 
is perpendicular to .
Parallel lines
An arbitrary segment
is drawn from point 
. Let 
be its intersection with 
. A circle centered on 
is drawn, and its intersections with 
and 
are 
. Likewise, draw a circle with the same radius centered on 
, and let its intersection with 
be 
. The distance between 
is measured with a compass, and the same distance is used to draw a circle from 
, thereby resulting in intersection 
. The line connecting 
is parallel to .
Division of a segment into equal parts
Draw a half line from the extremity
of a segment. Draw a circle centered on 
and let 
be the intersection with the half line. Using the same length 
, define 
on the half line. Connect 
through a line and draw lines parallel to 
passing through 
. The respective intersections are points that divide the segment into equal parts. 
Division of an angle into two equal halves
Draw a circle centered on
and let 
be its intersections with the half lines. Draw circles centered on 
and let 
be their intersection. The line connecting 
is the one that divides the angle into two equal angles.
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