Other Constructions

 Line Trisection We are given a line (AB), and we want to cut it into three equal pieces. To do this: Find M the midpoint of AB. Draw a circle at A and B (dotted), each with radius AB. They cross each other at C. At points A and B, draw a circle (purple) crossing M. At point M, draw a circle (cyan) crossing A. Circle M (cyan) intersects circles A and B (purple) at points D and E. Draw a line through points CD and CE (green dashed). CD and CE intersect AB at points G and H. AG, GH, and HB are all 1/3 of AB.
 Another method of line trisection doesn't involve that many circles, but rather a lot of lines! We are given segment AB (yellow): Draw a circle at A and a circle at B (dotted), each having a radius of AB. They cross each other at two points, C and D. Draw a line through points CA. CA (cyan) crosses circle A (dotted) at point E. Draw a line through points EB (red). Draw a line through points AD (magenta). AD (magenta) crosses EB (red) at point F. Draw a line through points CF. CF crosses AB at point G. AG is therefore 1/3 of AB!
 Line Division What if you needed to divide a line into an n number of parts, such as 5,7 or 11? The general construction for line division goes like this, given AB (yellow): Find any point P1 above AB. Draw a line through points A and P1. Find points P2...Pn where n is the number of segments you want to divide your line into (in this case,5). AP1=P1P2=...Pn-1Pn. Draw a line from Pn to B. Draw a line parallel to BPn from points Pn-1,Pn-2.... They cross AB at the points of n-section.
 Square Roots Using a compass and straightedge, it is even possible to construct the square root of any given line segment! We are given AB (yellow). In this example, it has a lenght of 3 and we want to construct the squareroot of 3. Extend AB. Find C on AB so that CA = 1. Find M, the midpoint of CB. Draw a circle at M (magenta) crossing C. From point A, draw a line perpendicular to it and find D, the intersection of that line with circle M (magenta). AD is therefore the squareroot of AB!
 Quadrature of a Rectangle The quadrature of a rectangle, is the construction of a square equal in area to a given rectangle. It is somewhat based on the construction for squareroots. Suppose we are given rectangle ABCD: Extend AB. Find E on AB so that BE = BC. Find M, the midpoint of AE. Draw a circle at M (cyan) crossing A. Draw a line from B (red) perpendicular to AB, crossing circle M (cyan) at N. BN is a side of the new square!
 Quadrature of a Triangle We already know how to square a rectangle. Then squaring a triangle should be no problem. Suppose we are given a triangle: Find the height of the triangle. Find a rectangle with equal area as the triangle (the rectangle has a height of half the height of the triangle and the same base). Square the rectangle.