How can you divide an angle into 3 equal angles? You may only use a straightedge and a compass to achieve this.

(This means : You have an angle A, you divide Angle A into 3 Angles B,C,D. And B=C=D=A/3)

Note: vohonam clarified that the problem actually only gives you a straightedge, not a ruler.

Open the compass at a random aperture, a small aperture would be fine. From the vertex of such angle, mark the lines of the angle sides 3 times, first from the vertex to the line, second from the point of intersection of first circle mark another circle of same radius, third mark another circle of same radius inntersecting the same line. Now open the compass from the angle's vertex to the first intersection of the line and mark the two lines, then open the compass from the vertex to the second point of intersection marked in the line and mark the two lines, finally open the compass from the vertex to the third point of intersection and mark the two lines. This process will make 3 lines perpendicular to the vertex and each one is bigger by 1 (the second line is twice the first and the third line is tree times the first). Now use the technique to find a perpendicular line in both points of the first line perpendicular to the vertex, and those perpendicular lines extend them on to the third line that is 3 times the first one. The last line that is perpendicular to the vertex, the one that is 3 times the first one, will be perfectly divided by 3, in those points of intersection just draw a line from the vertex to the two intersections in the line divided by 3 and you will get 3 equal angles of A/3. All of this shows the linear relation between the radius and any line perpendicular to the vertex that is opposite to the angle A

Posted by Antonio on 2003-09-04 06:15:00