A line parallel to one side
Draw a line parallel to a side. It cuts the other two sides in one ratio.
- Triangle ABC, with D on AB and E on AC.
- DE is drawn parallel to BC.
- AD is 4, DB is 6.
- AE is 6, EC is 9. Both splits read 2/3.
The theorem
Take a triangle ABC. Draw DE parallel to BC. D is on AB and E is on AC. The basic proportionality theorem says AD ÷ DB = AE ÷ EC. Here area(ADE) means the area of triangle ADE.
Draw a line parallel to one side of a triangle. It cuts the other two sides. It divides them in the same ratio.
The proof
Join B to E and C to D.
Triangles BDE and CDE share the base DE. B and C lie on BC, which is parallel to DE. So B and C are the same distance from DE. The two triangles have equal area. Area is half of base × height, and base and height are equal.
Triangles ADE and BDE have bases AD and DB on line AB. They have the same height, from E to AB. With equal heights, areas are in the ratio of the bases. So area(ADE) ÷ area(BDE) = AD ÷ DB.
In the same way, area(ADE) ÷ area(CDE) = AE ÷ EC. The bases are now AE and EC on the line AC.
The two areas we divide by are equal. They are area(BDE) and area(CDE). So AD ÷ DB = AE ÷ EC.
Slide D and watch the two ratios.
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