Английская Википедия:Harmonic quadrilateral

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Шаблон:Short description

In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle,[1] is a quadrilateral that can be inscribed in a circle (cyclic quadrangle) in which the products of the lengths of opposite sides are equal. It has several important properties.

Properties

Let Шаблон:Mvar be a harmonic quadrilateral and Шаблон:Mvar the midpoint of diagonal Шаблон:Mvar. Then:

Файл:Harmonic quadrilateral.svg
Tangents to the circumscribed circle at points Шаблон:Mvar and Шаблон:Mvar and the straight line Шаблон:Mvar either intersect at one point or are mutually parallel.
Файл:Harmonic quadrilateral2.svg
Angles Шаблон:Mvar and Шаблон:Mvar are equal.
Файл:Harmonic quadrilateral3.svg
The bisectors of the angles at Шаблон:Mvar and Шаблон:Mvar intersect on the diagonal Шаблон:Mvar.
Файл:Harmonic quadrilateral5.svg
The point of intersection of the diagonals is located towards the sides of the quadrilateral to proportional distances to the length of these sides.

References

Шаблон:Reflist

Further reading

  • Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913.

Шаблон:Polygons

Шаблон:Geometry-stub