Harmonic quadrilateral
Cyclic quadrilateral in which the products of opposite side lengths are equal
In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle,[1] is a quadrilateral that can be inscribed in a circle (cyclic quadrilateral) in which the products of the lengths of opposite sides are equal. It has several important properties.
Properties
Let ABCD be a harmonic quadrilateral and M the midpoint of diagonal AC. Then:
- Tangents to the circumscribed circle at points A and C and the straight line BD either intersect at one point or are mutually parallel.
- Angles ∠BMC and ∠DMC are equal.
- The bisectors of the angles at B and D intersect on the diagonal AC.
- A diagonal BD of the quadrilateral is a symmedian of the angles at B and D in the triangles ∆ABC and ∆ADC.
- 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
- ^ Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover, p. 100, ISBN 978-0-486-46237-0
Further reading
- Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913.
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Polygons (List)
- Antiparallelogram
- Bicentric
- Crossed
- Cyclic
- Equidiagonal
- Ex-tangential
- Harmonic
- Isosceles trapezoid
- Kite
- Orthodiagonal
- Parallelogram
- Rectangle
- Right kite
- Right trapezoid
- Rhombus
- Square
- Tangential
- Tangential trapezoid
- Trapezoid
of sides
1–10 sides | |
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11–20 sides | |
>20 sides |
- Pentagram
- Hexagram
- Heptagram
- Octagram
- Enneagram
- Decagram
- Hendecagram
- Dodecagram
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