Equilic Quadrilateral II: What is this about?
A Mathematical Droodle


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 http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Discussion

|Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny

The applet may suggest the following generalization of a theorem from [Honsberger] concerning equilic quadrilaterals:

Assume in a quadrilateral ABCD AD = BC, K, L, M, N are the midpoints of AC, AB, BD, AD. Then the quadrilateral KMLN is a rhombus.

(In [Honsberger] it is stated that, equilic ABCD, ΔKLN is equilateral. That ΔKMN is also quadrilateral is shown similarly. Putting the two together, KMLN is seen to be a rhombus. However, for it to be a rhombus, it is not necessary that the angle between AD and BC be 60°.)


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 http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The sides of the quadrilateral KMLN being the midlines in two pairs of triangles, one with base CD, the other base BC, the four sides are necessarily equal; hence also are pairwise parallel.

However, in an equilic quadrilateral, rhombus KLMN consists of two equilateral triangles.

References

  1. R. Honsberger, Mathematical Gems III, MAA, 1985, pp. 32-35

Equilic Quadrilateral

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

|Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny

 62049685

Search by google: