Download Notes on the Side-Splitting Theorem

Quadrilaterals in Euclidean Geometry
The side-splitting theorem
The Side-Splitting Theorem is an important result giving the proportions of segments in a triangle
cut by a line parallel to the base. In contrast to the Midpoint Connector Theorem, the line is not
necessarily passing through the midpoints of the sides.
We (finally) get some nice computational stuff in here, and I’ll focus on that. The theory leading up
to the Side-Splitting Theorem is the material on parallel projections. Please read it, and the proof of
this theorem (on pp 227-230), but it’s the end result of applying the theorem that we’ll look at here.
If a line parallel to the base BC of ABC cuts the other two sides AB and AC at points
E and F respectively, then
AF
AE
=
AB
AC
and
AE
AF
=
EB
FC
Example: In the triangle shown below, the following proportions could be set up. Note that there’s
not enough information here to solve for the unknowns.
30
x
=
70
y
30
x
=
x + 70
30 + y
70
y
=
x + 70
30 + y
70
x
=
30
y
Example: Given the triangle with lengths as marked, find x and y.
Solution:
Example: (don’t do this!) Although there is a proportional relationship between the marked side
lengths EF and BC, it is not the same as the other proportions. Do not ever set up the proportions shown on the screen and marked with a big red X.
Do this! What you need to do instead is drop in a parallel to one side at the other endpoint of the
segment (e.g. a parallel to AB at point F ):
Since (by construction), F G||EB, BEF G is a parallelogram with congruent opposite sides, and you
can transfer the length EF down (shown as s). This breaks the base into lengths s and t − s.
This allows you to look at the triangle “sideways” and apply the side splitting theorem with F G as
the splitter.
x
s
=
z
t−s
s
x
=
x+z
t
z
t−s
=
x+z
t
An example with numbers: Various lengths in the diagram shown:
Drop the parallel down.
(Write down the calculations given in the lecture for this example.)
Example: Given the triangle with lengths as marked, find x and y.
Solution: