[alogo] Quandrangle division

Here is the answer to a problem, proposed by Michael Metaxas: Given a convex quadrangle, divide it through two intersecting lines in four equal parts (in area). Following an idea of Michael Papadimitrakis, find first the envelope of the lines [EF] dividing the quadrangle in two equal parts. As pointed out by Antreas Varverakis, this envelope is a hyperbola, with asymptotic lines [AB] and [CD] (drawn in red), or [AD] and [BC] (not drawn). Then draw a tangent [EF] to this hyperbola at a point H of it. This creates (under certain restrictions) two other quadrangles: AEID and BEIC, having equal areas. Repeat the construction of the envelope for these two quadrangles. This determines two other hyperbolas (green and blue respectively). Draw a common tangent [FG] to these two hyperbolas. This completes the construction of the two lines, intersecting at F. Obviously there are infinite many solutions.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]

The fact that the envelope of the lines, dividing in two equal parts the quadrangle, is a hyperbola, is due to the fact that asymptotic triangles of a hyperbola have a constant area. Look at the file AsymptoticTriangle.html for a picture.

Look at the file ParaDivision.html for the solution of the special problem of a four-division of a parallelogram. In that case point F is always the center of the parallelogram. Which is the locus of F in the general case?

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