The first part of the week will be devoted to area and some more ratios and coordinates. Friday will begin some work on polyhedra. Monday of the next week is the midterm.

Read about area B&B Chapter 7 and Heilbron, Chapter 3. Also read about Ceva's theorem in that chapter of Heilbron.

Draw a triangle ABC. It should be of medium size on the page.

The general question is this. If x, y, z are any numbers so that x + y + z = 1, find a point P so that the barycentric coordinates of P are x, y, z.

In other words, you will be told (x, y, z); find the point P that goes with these values of x, y, z. You can use a marked ruler for this. You don't have to construct with unmarked straightedge and compass unless you want to. Or you can use Sketchpad.

The point P in the figure will not be inside the triangle for some values of x, y, z.

Hint: You know how to find the barycentric coordinates of P using lines through P parallel to the sides of ABC. Reverse this process.

Problem: Draw your triangle ABC and the draw and label the following points that correspond to the given numbers (x, y, z).

1. Point P, where (x,y,z) = (1/3, 1/3, 1/3).
2. Point Q, where (x,y,z) = (1/2, 1/4, 1/4).
3. Point R, where (x,y,z) = (1/2, 1/2, 0).
4. Point S, where (x,y,z) = (0, 1/4, 3/4).
5. Point T, where (x,y,z) = (0, 1, 0).
6. Point U, where (x,y,z) = (2, 1, -2).
7. Point V, where (x,y,z) = (1, 1, -1)).
8. Point W, where (x,y,z) = (1, 0, -1).

Given a rectangle ABCD with sides 13 units = |AB| and 5units =|BC|. Let E be the point on AB at distance 5 from B and G be the point on CD at distance 5 from D. Construct the perpendiculars to the sides at E and G. Let these perpendiculars intersect the diagonal AC at points F and H.

Then the rectangle can be cut into the two triangles and two quadrilaterals as in the figure.

Imagine reassembling these four pieces into the square of the second figure.

The area of the rectangle is 65. The area of the square is 64. What happened to the extra square unit?

Given a parallelogram ABCD. Let M be the midpoint of CD and N be the midpoint of BC. If E is the intersection of BD with AM and F is the intersection of BD with AN, prove that E and F divide the segment BD into 3 equal parts.

Prove that the interior angle bisectors of a rectangle intersect in four points which are the vertices of a square. What about the exterior angle bisectors?

Given a parallelogram ABCD. If E is a point on the diagonal BD, the lines through E parallel to the sides cut the parallelogram ABCD into four smaller parallelograms as shown in the figure. Prove that the shaded parallelograms AIEH and CGEF have the same area.

Let ABC be a triangle and let D and E be points on the sides of the triangle so that DE is parallel to BC as in the figure.

Give reasons for all your answers.

1. If F is a point on segment DE, then prove that triangle BCF has the same area as triangle BCD, no matter where on the segment DE that F is located.
2. If the ratio |AD|/|AB| = k and the area of triangle ABC is T, what is the area of BCF?
3. Let G be the intersection of line AF with BC. Show that the ratio

(area BGA)/(area GCA) = (area BGF)/(area GCF) = (area BFA)/(area FCA)

4. If the barycentric coordinates of F with respect to ABC are (1/6, 1/2, 1/3) and the area of triangle ABC is 12, what is the area of triangle BCF? What are the areas of triangle CAF and ABF?

Suppose r, s and t are constants. Then show how to solve this system of equations for x, y, z when rst = 1:

r = x/y; s = y/z: t = z/x.

Explain why rst = 1 is necessary. Work out the answer for this numerical case: r = 2/3. s = 3/8, t = 4.