harmonic mean in trapezoid

Theorem.  If a line parallelMathworldPlanetmathPlanetmath to the bases of a trapezoidMathworldPlanetmath passes through the intersecting point of the diagonals, then the portion of the line inside the trapezoid is the harmonic mean of the bases.

Proof.  Let AB and DC be the bases of a trapezoid ABCD and E the intersecting point of the diagonals of ABCD. Denote the cutting point of AD and the line through E and parallel to the bases by P, and the cutting point of BC and the same line by Q.  Then we have


with line ratiokh=CDAB, where h and k are the heights of the triangles ABE and CDE, respectively, when h+k equals the height of the trapezoid.  We have also


with line ratio


Thus we can express the length of PE as


Similarly we may determine EQ and that  EQ=PE.  Consequently,


which is the harmonic mean of the bases AB and CD.

Title harmonic mean in trapezoid
Canonical name HarmonicMeanInTrapezoid
Date of creation 2013-03-22 17:49:22
Last modified on 2013-03-22 17:49:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 26B99
Classification msc 51M04
Classification msc 51M15
Related topic HarmonicMean
Related topic SimilarityOfTriangles
Related topic CorrespondingAnglesInTransversalCutting
Related topic SimilarityInGeometry
Related topic MedianOfTrapezoid
Related topic ConstructionOfContraharmonicMeanOfTwoSegments
Related topic IntegerHarmonicMeans