The third fallacious proof is:

Consider triangle ABC, with MN parallel to BC and MN intersecting AB and AC in poins M and N, respecitvely.

We will now prove that BC = MN.

Because MN is parallel to BC, we have triangle AMN ~ triangle

ABC and BC/MN = AB/MN.

It then follows that:

Now, multiply both sides of this equality by (BC - MN) to get:

By adding (BC)(AM)(MN) - (AB)(MN)(BC) to both sides, we get:

This equation can be written as:

No discussion of mathematical fallacies would
be complete without an example of a dilemma from division by __zero__.
We committed this mathematical sin when we divided by zero in
the form of [(BC)(AM) - (AB)(MN)], which was a consequence of
the triangles proved to be similar earlier.