[alogo] 1. Trapezium's basic properties

1) The non-parallel sides extended cut at point G, through which passes line EF,
     joining the middles of the parallel sides.
2) On EF lies also the intersection point H of the diagonals of the trapezium.
3) Points {E,F,H,G} form a harmonic tetrad (or division see Harmonic.html ).
    Last property holds also for general quadrangles (see Quadrangle_0.html ).


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1) Draw the medial line GE of the triangle GDC and show that it is also medial to triangle GAB.
2) Use Ceva's theorem (see Ceva.html ).
3) HE/HF = DE/FB, GE/GF = DE/AF etc.

In the file ParallelMedians.html we discuss an application of the previous properties to triangles.

[alogo] 2. Metric relations

1) The diagonals  d1=AC,  d2 = BD  and and sides as shown in the figure satisfy [Rouche, p. 329]:
                                                         d12 + d22 = c2 + d2 +2ab.
2)   It is also [Rouche, p. 330]:
                                                         (d22 - d12)/(d2 - c2) = (a+b)/(b-a).
To prove (1) write the four following equations combine them to eliminate cosines and divide
the result by (a+b):

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To show the second property do a similar calculation using triangles AHD and BHC,
setting AH = kd1, BH = kd2, HC = k'd1, HD = k'd2, with k/k' = b/a (see section-1).


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[alogo] 3. Trapezium construction

To construct a trapezium knowing its two diagonals d1, d2 and its two non-parallel sides
d and c [Rouche, p. 335].

Combine the two equations of section-2 involving also {a,b} to find these lengths. Then
construct the trapezium knowing {a,b,c,d,d1,d2}.

Some additional properties of trapezia are discussed in the files Trapezium.html , Trapezium2.html .

See Also

Harmonic.html
Quadrangle_0.html
Ceva.html
ParallelMedians.html
Trapezium.html
Trapezium2.html

Bibliography

[Rouche] Rouche, Eugene et Comberouse Ch. Traite de Geometrie elementaire Gauthier-Villars, Paris 1873

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