Some Ratios in a Triangle

Draw a triangle ABC and a point P inside the triangle. Denote the area of a triangle such as ABC by <ABC>.

• Compute the areas of the triangles ABC and PBC; then compute the ratio x = <PBC>/<ABC>.

• Now construct the line AP and the intersection P1 of AP and BC.
• Also construct the line through P parallel to BC and the intersections M and N of this line with AB and AC, respectively.

Now compute the following ratios and explain any ratios that you find that are equal to x or to 1 – x.

Also construct altitude segments PP' and AA' perpendicular to BC, where P' and A' are on line BC.

• Compute PP'/AA'.  What happens to the ratio if angle C is obtuse?

Lines Parallel to BC

Construct a line q parallel to BC through a free point Q.  Hide the line MN and points M and N.   Merge point P to line q.  Then answer these questions.

• When q is fixed, does x stay the same as P moves along q?  If so, why?
• For what position of q is the value of x equal to 1?
• State an interpretation of x as a relative A-height or A-level of P.

Multiple ratios in a triangle

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

• Compute the areas of the 4 triangles ABC, PBC, PCA, PAB.
• Then compute the ratios of the areas: 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 6C)

1. If the side lengths of ABC are a = BC, b = CA, c = AB, then using only x, y, z and a, b, c, write down the lengths of every segment in the figure.
2. The three shaded triangles are similar to triangle ABC, what is the scaling factor (ratio of similitude) in each case.
3. If the area of triangle ABC is T, what are the areas of the 3 triangles and the 3 quadrilaterals into which ABC is dissected in the figure?  Use algebra to show that the 6 areas add up to the area of ABC.
4. P divides each parallel segment.  Tell these 3 ratios, using x, y, z, a, b, c: PC2/PB1, PA2/PC1, PA1/PB2.
5. 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?
6. For what points in the triangle is x = 0?
7. For what points in the triangle is y = z?
8. If x = y = z, where is P?