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Point Incidence Geometry Diagonal Basic Length Triangle Similarity In Geometry Parallellism In Euclidean Plane Prismatoid Trapezoid Height Of A Triangle Harmonic Mean

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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 and be the bases of a trapezoid and the intersecting point of the diagonals of . Denote the cutting point of and the line through and parallel to the bases by , and the cutting point of and the same line by . Then we have

$\displaystyle \Delta CDE \sim \Delta ABE$

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

and

are the heights of the triangles

and

, respectively, when $h\!+\!k$ equals the height of the trapezoid. We have also

$\displaystyle \Delta PED \sim \Delta ABD$

with line ratio

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

Thus we can express the length of

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

Similarly we may determine

and state that

. Consequently,

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

which is the harmonic mean of the bases

and

$\begin{pspicture}(-1,-1)(7,4) \pspolygon(0,0)(6,0)(5,4)(2,4) \psline[linecolor=b... ...3.45){$k$} \psline(3.33,0.2)(3.11,0.2) \psline(3.11,0.2)(3.11,0) \end{pspicture}$

1	Diagonal
1	Incidence Geometry
1	Triangle
1	Point
1	Similarity In Geometry
1	Basic Length
2	Parallellism In Euclidean Plane
2	Prismatoid
3	Trapezoid
4	Height Of A Triangle
6	Harmonic Mean