Dividing Any Line into 3rds or 5ths Using a Triangle.

 I discovered this method to divide any line into 3 or 5 equal sections by just playing around with the drafting features of my CNC software.  It has come in handy many times while creating joinery for furniture connections.  Sizing a tenon for the end of a stretcher there is a 1/3 rule which suggests the tenon thickness should be 1/3 the thickness of the stretcher.  When I'm connecting a narrow board into a wider board I like to increase the relative size of the tenon to 3/5ths of the stretcher thickness. One 5th for the perimeter shoulder and three 5ths for the tenon.  

I haven't seen this method used anywhere on the web or in print.  It may exist out there some where.  When I look for methods to divide any line by 3 or 5 equal sections I find most use an adjacent connected line with marked off known segments, projected parallel across to the starting vector to divide it.  I used that trick when drafting was done with a parallel bar, triangles, compass and straight edges. There is another easy trick to dividing the width of a board using a ruler/scale held diagonally across the board at marks equal to the number of divisions you want. 

This triangle method starts by adding two lines to your starting vector to make a triangle. Now find the center points of all three sides of the triangle.  This works with any size or shape of triangle, although the density of lines created is easier to draw and see on a large triangle closer to equilateral. Be accurate.  Mistakes here propagate through following steps. 

Find Centers of Each Side.

Next connect the centers to each other.

Connect Centers

Now draw in the medians by connecting the corners to the opposing center points.

Corners to Centers

You can see where each center line intersects the center triangle lines.  If you draw a line from a corner through the opposing intersections, it will cross the opposing face at 1/3 over from a corner. 

Thirds, in Red

Do this for all 3 sides.

3rds on all 3 sides.

The new vectors intersect the previous vectors, and provide useful nodes we can use to divide each side into 5 equal segments. 

Project to find 5ths, in Green.

It is easy to get confused about which intersections to use.  Basically for each face there is a red/red, a red/black. a red/black. and a red/red intersection to pass through. 

5 Equal Segments.

I usually only need to go far enough to find 5ths on one side, but it is easy to project the other two sides to divide them both into 5 segments.

5ths on all 3 Sides.

There are further divisions that can be found with the new intersections created. I rarely need 7ths or 9ths though and have a much more automated way of dividing line segments in my CNC software.  

For example:

7ths. In Orange. 

9ths. In Purple

I suspect this method will work for odd numbers up to infinity so long as you have infinite patience and a very very fine pencil. I extended it to 11ths with no trouble but the density of vectors is making it harder with each step up.

11ths

Of course since 13 is a lucky number I went ahead and proved that 13 segments of a vector can also be found. 

13ths.  Just Because.

I dedicate this post to my Sister Vickie's husband Chris Z. A useful application of geometry. 3rds and 5ths (and 7 and 9 and 11 and 13), with no parallel lines needed.  If no one else before me has discovered this method, I name it the D.A.Brown Method. Published here first on 7/9/2023. 

As a final teaser, you can also divide the triangle sides by even numbered segments using a variation of projected lines through intersections of previous vectors used.   Although this works with any shape of triangles, you can also do the same with 4 sided polygons. With rectangles, trapezoids, or parallelograms this is easiest to see. How about 5 sided polygons? 

4D