Shades....most of this is straight out of Euclid's Elements......Book VI....Proposition 2
Here it is: http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI2.html
I won't repeat it verbatim, because I can't do a better job than Eulclid....LOL!!!!
You need to use the second part under the long dashed line that begins with...
"Next, let the sides AB and AC of the triangle ABC be cut proportionally, so that BD is to AD as CE is to AE. Join DE."
At the end of the proof....it is shown that the lines are parallel.....now....all we need to add is this :
m<ADE = m<ABC the corresponding angles formed by a transversal [in this case, the transversal AB] cutting two parallel lines are congruent
Thus, by AA congruency.....triangle ADE ≈ triangle ABC
And.....similar triangles are similar in all respects......thus AD is half of AB....therefore....DE is half of BC [ thus....the line segment joining the sides = 1/2 of the base]
Let me know if you get stuck......I'll see if I can help you out....!!!!
