Triangle Proportionality Theorem

Press on the numbers to see the steps of the proof.

1

Draw \(\small\Delta\mathtt{ABC}\). Then draw \(\mathtt{\overline{DE}}\) inside the triangle so that it is parallel with \(\mathtt{\overline{BC}}\).

2

Line segments \(\mathtt{AB}\) and \(\mathtt{AC}\) form transversals to the parallel lines \(\mathtt{DE}\) and \(\mathtt{BC}\).

3

Corresponding angles are congruent.

\(\small\Delta\mathtt{ABC}\) is similar to \(\small\Delta\mathtt{ADE}\) by AA Similarity.

\(\small\Delta\mathtt{ABC}\) is similar to \(\small\Delta\mathtt{ADE}\) by AA Similarity.

4

\(\mathtt{\frac{\color{blue}{BA}}{\color{red}{DA}} = \frac{\color{purple}{CA}}{\color{orange}{EA}}}\), so \(\mathtt{\frac{\color{blue}{a + b}}{\color{red}{b}} = \frac{\color{purple}{c + d}}{\color{orange}{d}}}\).

Rewrite this as \(\mathtt{\frac{\color{red}{b}}{\color{red}{b}} + \frac{a}{\color{red}{b}} = \frac{\color{orange}{d}}{\color{orange}{d}} + \frac{c}{\color{orange}{d}}}\).

5

The fractions that equal 1 cancel, so \(\mathtt{\frac{a}{\color{red}{b}} = \frac{c}{\color{orange}{d}}}\).

The segment \(\mathtt{DE}\) divides \(\mathtt{\overline{\color{blue}{AB}}}\) and \(\mathtt{\overline{\color{purple}{AC}}}\) proportionally.

The segment \(\mathtt{DE}\) divides \(\mathtt{\overline{\color{blue}{AB}}}\) and \(\mathtt{\overline{\color{purple}{AC}}}\) proportionally.

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