A theorem that is credited to Thales (after Thales of Miletus, (c. 620 - c. 546 BC) is the subject of Euclid's Elements VI.2:
If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the other sides of the triangle, or these produced, proportionally; and, if the sides of the triangle, or the sides produced, are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle.
(The applet below illustrates the theorem and its proof. Points A, B, C, D are draggable.)
24 December 2013, Created with GeoGebra
Given DE||BC, we want to prove that BD/AD = CE/AE.
Join BE, CD. Then triangles BDE and CDE have equal areas by Elements I.38:
Turning to triangles CDE and ADE, we similarly observe that
AD/BD = AE/CE,
Now for the converse. Given AD/BD = AE/CE, prove that DE||BC.
For a proof, note that the steps above are reversible. First we have
Note that AD/BD = AC/CE is equivalent to AB/BD = AC/CE. In particular, triangles ABC and ADE are similar so that their bases are in the same ratio:
AB/BD = AC/CE = BC/DE.
As a consequence, if DE is a midline of ΔABC, i.e., if AD = BD and
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