# Equilateral Triangles on Diagonals of Antiequilic Quadrilateral: What is this about? A Mathematical Droodle

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Discussion

Playing with the applet that demonstrates some properties of equilic quadrilaterals it is not hard to observe that the quadrilateral with two equal opposite diagonals inclined to each other at 120° also has interesting properties. As an HTML page needs to have a title relevant to its content, such a quadrilateral appeared to beg for a name. In deference to the origins of the shape I have named the quadrilateral antiequilic. Let ABCD be such an antiequilic quadrilateral (AD = BC).

If equilateral triangles ACP and BDR are drawn on the diagonals AC and DB (towards AB) of an antiequilic quadrilateral ABCD, then their apexes P and R coincide. In addition, this point lies on the bisector of angle ASB, where S is the intersection of AD and BC.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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3. Equilateral Triangles on Segments of Equilic Quadrilateral
4. Equilic Quadrilateral I, A Variation
5. Equilateral Triangles on Diagonals of Antiequilic Quadrilateral

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