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Ratio of Segment Lengths Between Parallel Lines

Ratio of Segment Lengths Between Parallel Lines in a Triangle

, let a line parallel to

intersect

and

at points

and

, respectively.

and

Thus,

Since the ratios of the corresponding side lengths in similar triangles are equal:

If a line passing through point

and parallel to

intersects

at point

, then in

and

Thus, the ratios of the corresponding side lengths are equal:

Since

is a parallelogram,

. Therefore:

The same result holds even if points

and

are on the extensions of

and

, respectively.

Properties of Segment Ratios in a Triangle

, if points

and

lie on

and

(or their extensions), and

Ratio of Segment Lengths Between Parallel Lines

Consider three parallel lines

, and

intersecting two other lines

and

at points

and

, respectively.

If a line passing through

and parallel to

intersects

and

at points

and

, then in

, since

Also, since

and

are parallelograms:

Thus:

Properties of Segment Ratios Between Parallel Lines

When three parallel lines intersect two other lines, the ratios of the segment lengths between the parallel lines are equal.
In the following figure, if the three parallel lines divide the lines

and

into segments

, and