The three parallel lines theorem is another theorem that provides a ratio between line segments created by a transversal of parallel lines, similar to the Intercept Theorem.
It states that if three lines that are all parallel to each other are intersected by two transversal lines, the line segments of the traversal lines cut by the parallel line have equal proportions:
|AB|/|BC|=|DE|/|EF|
Problem
Two transversal lines intersect three parallel lines, x, y, and z, at points A,B,C and D, E,F respectively. Show that |AB|/|BC|=|DE|/|EF|
Strategy
This theorem is very similar to the Intercept Theorem, and the drawing looks very much like that of the intercept theorem:
So let’s see if we can use this theorem in our proof. We’ll draw a line parallel to DF through point A (Remember the axiom: “for every straight line and every point not on that line, there is one straight line that passes through that point, parallel to the first line.”) :
Now the left side of the drawing is exactly what we have in the intercept theorem. Using that, directly, we have |AB|/|BC|=|AG|/|GH|.
But as x,y, and z are all parallel, and AH is parallel to DF by construction, ADEG and GEFH are both parallelograms. And in a parallelogram, opposite sides are equal, so |AG|=|DE| and |GH|=|EF|. Substituting, we have the needed result: |AB|/|BC|=|AG|/|GH|=|DE|/|EF|
And this is how we prove the Three Parallel Lines Theorem.
Proof
(1) x || y || z //Given
(2) AH||DF //Construction
(3) |AB|/|BC|=|AG|/|GH| //Intercept theorem
(4) |AG|=|DE| //(1), (2), Opposite sides of a parallelogram are equal
(5) |GH|=|EF| //(1), (2), Opposite sides of a parallelogram are equal
(6) |AB|/|BC|=|DE|/|EF| //(3), (4), (5) , Transitive property if equality
