Introduction Deductive reasoning has always been an important part of a scientists or engineer's collection of abilities. From developing new theoretical results to problem identification and solution, thinking in sound, logical steps is a key element. Historically, the place where people are grounded in this is plane geometry. In carrying out proofs of properties from plane geometry, one not only learns some useful geometry but also develops important analytical thinking capabilities.

Background vectors; Kolman 3.1, Bretscher Appendix A

In the 9 problems that follow, prove each of the following using vector algebra only unless otherwise stated. Do not resort to breaking vectors down into components or coordinates (except for 6b). Clearly state the steps used in establishing each proof. No proofs by example! Supporting sketches appreciated.

1. For an arbitrary triangle, T, if one connects the midpoints of any two sides, the resulting line segment is parallel to the third side and half of its length.

2. Let A, B, C, and D be the vertices of a quadrilateral. Let m₁, m₂, m₃, and m₄ be the midpoints of the sides AB, BC, CD and DA in that order. Prove that the figure formed by connnecting m₁ m₂ m₃ m₄ is a parallelogram.

3.a. Show the diagonals of a parallelogram bisect each other

b. Show that the diagonals are orthogonal if and only if the sides are of equal length

c. Show the diagonals are of equal length if and only if the parallelogram is a rectangle

d. Show , where x and y are vectors, that

|| x + y ||² + || x - y ||² = 2|| x ||² + 2|| y ||²

and interpret this result geometrically.

4. a. Use the result from 3d to establish the identity

x y = 1/4 (|| x + y ||² - || x - y ||²)

b. Use part a to conclude that a parallelogram is a rectangle if and only if its diagonals are of equal length.

5. Consider an arbitrary triangle. Use vector techniques to show that the sum of the vectors from the midpoint of each side to the opposite vertex is 0.

6. a. Prove that the medians of a triangle intersect at a single point. (define median first.)

b.This point of intersection is called the centroid of the triangle. If the three points defining the triangle are (x₁,y₁), (x₂,y₂), and (x₃,y₃) then the centroid has coordinates

(x₁ + x₂ + x₃, y₁ + y₂+ y₃)/3

Supply at least one visual example with actual coordinates to demonstrate this problem.

7. If ABCD is an arbitray parallelogram, prove that the line joining A to the midpoint of CD intersects the diagonal BD in one of the trisection points of the diagonal.

8. Use vector methods to show that the altitudes of an arbitrary triangle meet at a common point.

9. If x and y are nonzero vectors, find a scalar c such that the vector h = y - cx is orthogonal      to x. It is given by the formula

                                c = x* y/x* x                     (* indicates dot products)

       provide an example to demonstrate this result.

b. If x and y are thought to determine a parallelogram, interpret h geometrically.

c. Use parts a and b to write an expression for the area of the parallelogram determined by

     x and y purely in terms of x and y. Provide an example demonstrating this result.