Lab 01 Part C: Multiple ratios in a triangle

The answers to the questions here will be part of Assignment 1. It is hoped that the lab will help in answering them.

Make a new page with the same triangle ABC and P in the triangle.

• Denote the side lengths of ABC by a = BC, b = CA, c = AB.
• Compute the coordinates x, y, z of P as before, i.e, if <ABC> denotes signed area of a triangle, then x = <PBC>/<ABC>, y = <PCA>/<ABC>, z = <PAB>/<ABC>.

Construct 3 lines through P parallel to the 3 sides of the triangle.  Label the sides as in the figure.

Problems (these will be part of Assignment 1)

1. On a separate sheet of paper, write down the lengths of every segment in the figure in terms of a, b, c, and x, y, z. Use Sketchpad to confirm your work.
2. The three shaded triangles are similar to triangle ABC, what is the scaling factor (ratio of similitude) in each case? How do you know this?
3. If the area of triangle ABC is T, what are the areas of each of the 3 triangles and the 3 quadrilaterals into which ABC is dissected in the figure?  The answers should be in terms of a, b, c, x, y, z and T.
4. Use algebra to show that the 6 areas add up to the area of ABC.
5. P divides each of the 3 parallel segments.  Write down these 3 ratios (using x, y, z, a, b, c): PC2/PB1, PA2/PC1, PA1/PB2.
6. Let the lines AP, BP, CP intersect the opposite sides of the triangle ABC in points A', B', C'.  What are the ratios A'B/A'C, B'C/B'A, C'A/C'B?