intercept theorem

Theorem.  If two intersecting lines are cut by parallel lines, the line segments cut by the parallel lines from one of the lines are proportional to the corresponding line segments cut by them from the other line.

The theorem may be condensed to the following form:

• If a line parallel to a side (http://planetmath.org/Triangle) $BC$ of a triangle $ABC$ intersects the other sides in the points $D$ and $E$, then the proportion equation

  $BD:DA=CE:EA$ (1)

  is true.

The intercept theorem has been known by the ancient Babylonians and Egyptians, but the first known proof is found in Euclid’s Elements.

Proof.  The areas of triangles, which have equal heights, are proportional to the bases of the triangles; if the bases are equal, then also the areas are equal.  These facts are used in the

$$BD:DA=\mathrm{\Delta }BDE:\mathrm{\Delta }DAE=\mathrm{\Delta }CED:\mathrm{\Delta }EAD=CE:EA$$

of equalities. Q.E.D.

Title	intercept theorem
Canonical name	InterceptTheorem
Date of creation	2013-03-22 18:49:42
Last modified on	2013-03-22 18:49:42
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51-01
Classification	msc 51M04
Related topic	AreaOfAPolygonalRegion
Related topic	SimilarTriangles
Related topic	MidSegmentTheorem