# 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 $\frac{k}{h}}={\displaystyle \frac{CD}{AB}$, 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 |