Part 4. Introduction to Affine Coordinates

Let's return to the picture of a triangle ABC and an additional point D, which we wish to locate using affine ratios.  For an affine transformation T that takes ABC to A'B'C', we can use these affine ratios to locate the image D'.

In Part 2, we suggested an approach using diagonals.

In this section, we will look at another approach based on parallel lines.

• Construct a triangle ABC with lines as extended sides. 
• Then construct any point D and the 2 lines through D parallel to AB and AC. 
• Let the intersections D1 and D2 be as in the figure. 
• Now on AB and AC the points D1 and D2 can be located by signed ratios.  AD1/AB and AD2/AC.  Measure these ratios.

Construction

Given a triangle A'B'C', use these ratios to find D.

Does this kind of description of the location of D look familiar?  What if AB and AC are perpendicular?

Another collection of ratios

• Add to the figure the line through D parallel to BC. 

• Then each side of triangle ABC is cut into 3 parts.  For each side measure the 3 ratios of the 3 segments to the length of the corresponding side. 
• Do you find a relationship among the ratios for each side?  What is it?