Triangle Proportionality Theorem
Lesson · Beginner
Geometry: Plane Geometry
Triangle Proportionality Theorem
Equivalently,
Converse of the Triangle Proportionality Theorem
Equivalently,
Since DE ∥ BC,
Therefore, by AA similarity,
For the converse,
By Ratio Manipulation,
Therefore, by SAS similarity,
Therefore,
A line through the midpoint of one side of a triangle, parallel to a second side, bisects the third side.
If D is the midpoint of AB and DE ∥ BC, then
so
Therefore, E is the midpoint of AC.
Also,
so
The segment joining the midpoints of two sides of a triangle is parallel to the third side and has half its length.
AM is a median of triangle ABC, and G is its centroid (the intersection point of the medians). Prove that AG : GM = 2 : 1.
By AA,
By (1),
Similarly,
In triangle ABC, points D, F lie on AB in the order A − D − F − B, and points E, G lie on AC in the order A − E − G − C. Suppose DE ∥ FG ∥ BC, [ADE] = 25, [DFGE] = 56, and FB = 2AD. Find [FGCB].
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