A line is projected to a line.
Parallel lines are projected to parallel lines.
If A, B, C are collinear points that are projected to A’, B’, C’, then the ratio AB/BC = A’B’/B’C’.
The following transformations can be obtained as composition of parallel projections.
Translations
Strains (hence dilations) [This follows from Monday’s homework.]
Shears
Let O be a point in the plane, with A and B points distinct from O. If A’ is a point on OA (distinct from O) and B’ is a point on OB with (signed) ratio OA’/OA = OB’/OB, then line A’B’ is parallel to line AB.
Corollary. If ABC is a triangle, and A’, B’, C’ are the midpoints of the sides opposite A, B, C, then the sides of the triangle A’B’C’ are parallel to the respective sides of ABC.
Assignment for Wednesday, 10/10 (most work done in groups in class)
Problem 1. Prove that for any quadrilateral ABCD, the quadrilateral whose vertices are the midpoints of the sides of ABCD is a parallelogram.
Problem 2. Prove that the diagonals of a parallelogram bisect each other. (already done in first class).
Problem 3. Given a triangle ABC, let the midpoints B’ and C’ be the midpoints of sides CA and AB. Then let G be the intersection of lines BB’ and CC’. Finally, let B’’ be the midpoint of BG and C’’ be the midpoint of CG. Draw this figure and use a parallelogram in the figure to prove that the ratio of GB/GB’ = GC/GC’. What is this ratio?