harmonic mean in trapezoid

Theorem.  If a line parallel to the bases of a trapezoid 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

$$\mathrm{\Delta }CDE\sim \mathrm{\Delta }ABE$$

with line ratio  ${\displaystyle \frac{k}{h}}={\displaystyle \frac{CD}{AB}}$, 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

$$\mathrm{\Delta }PED\sim \mathrm{\Delta }ABD$$

with line ratio

$$PE:AB=\frac{k}{h+k}=\frac{\frac{k}{h}}{1+\frac{k}{h}}=\frac{\frac{CD}{AB}}{1+\frac{CD}{AB}}.$$

Thus we can express the length of $PE$ as

$$PE=AB\cdot \frac{\frac{CD}{AB}}{1+\frac{CD}{AB}}=\frac{CD}{1+\frac{CD}{AB}}=\frac{AB\cdot CD}{AB+CD}.$$

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

$$PQ=PE+EQ=\frac{2\cdot AB\cdot CD}{AB+CD},$$

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