Triangle Proportionality Theorem Calculator

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. In triangle $ABC$ with $DE \parallel BC$:

$$\frac{AB}{AD} = \frac{AC}{AE} = \frac{BC}{DE}$$

Worked Example

If $AB = 12$, $AD = 4$, and $AC = 15$, then $AE = \frac{AC \times AD}{AB} = \frac{15 \times 4}{12} = 5$. Also $DB = AB - AD = 8$ and $EC = AC - AE = 10$.

Frequently Asked Questions

What does the converse state?

If a line divides two sides of a triangle proportionally, then the line is parallel to the third side.

How do I find DE when DE is parallel to BC?

Use $\frac{AB}{AD} = \frac{BC}{DE}$, so $DE = \frac{BC \times AD}{AB}$.

Why must AD be less than AB?

Point D must lie between A and B on side AB. If AD equals or exceeds AB, the segment setup is invalid.

Is this the same as the midsegment theorem?

The midsegment is a special case where D and E are midpoints, giving $DE = \frac{BC}{2}$.

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Triangle Proportionality Theorem Calculator