# 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

 $\Delta CDE\;\sim\;\Delta ABE$

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

 $\Delta PED\;\sim\;\Delta ABD$

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