Examples
Basic Examples (3)
Find three harmonic quadrilaterals that contain a triangle:
Show these:
The products of the lengths of opposite sides are equal:
Scope (3)
Some spiral similar triangles pairs (marked as red and blue) can be found by halving the diagonal or the extension of a side:
Take the midpoint of one diagonal:
Double the length of one side:
Neat Examples (3)
The vertices of a harmonic quadrilateral are inverse to some squares and vice versa. To find a proper inversion center, use the intersection of the orthogonal circles passing through the vertices of each diagonal:
The intersections of the two solid orthogonal circles are the proper inversion centers:
Either one can be chosen as the inversion center. The gray hatch-filled polygon is a square for each diagram:
A weaker version of the result is that any cyclic quadrilateral (not necessary harmonic) is the inversion of some rectangle. The inversion center is found in the similar way as described above.
Generate a set of random harmonic quadrilaterals:
Check the distribution of the area of harmonic quadrilaterals in the unit circle:
Related Links
Version History
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1.0.1
– 04 May 2022
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1.0.0
– 25 April 2022
Related Resources
Author Notes
This function provides useful features to explorer the corresponding chapters in Advanced Euclidean Geometry and Euclidean Geometry in Mathematical Olympiads.