To start, my apologies for not having a better image for this.
A simple way to draw a 45º angle from a point on a line was shown in the book By Hand & Eye by Jim Tolpin & George R. Walker.
Simple 45º Angle Construction.jpg
The only tools required are; a straight edge, compass and pencil.
From the point of origin “O” a radius is struck to produce point P1 and P2 on the line. a dashed circle was drawn to indicate this radius.
From P1 and P2 using a larger radius, two arcs are drawn to intersect above point O. This is how to drop a line from R1 to O creating a right angle (90º). In this case a full circle was drawn. Using the same radius an arc is drawn from R1 to R2. The arc at R3 was drawn from R2 to show a center line from P2. As is commonly known continuing around the circle with its radius produces the layout for a hexagon, I am sure this is part of the reason this works.
A line drawn from point O to the point at R2 creates a 45º angle from point O on the line formed from P1 to P2.
Since varying the radiuses results in the same orientation of the hexagon in relation to line R1 - O and P1 - P2 it seems this plays a part in why a 45º angle results when a line is drawn from O to R2.
I am curious as to why this works.
Anyone know the reason?
jtk