# A Quadrilateral with 3 Equal Sides

What is this about?

A Mathematical Droodle

What if applet does not run? |

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Copyright © 1996-2018 Alexander BogomolnyThe applet lets you manipulate a quadrilateral, ABCD, so as to maintain the equality of three sides:

BC = CD = AD.

Let E be the point of intersection of the diagonals. An interesting situation occurs when also

Clearly, in such cases, CE ≠ DE, for, otherwise, the quadrilateral would be a trapezoid. Interestingly, when this condition holds, invariably

Following M. Hajja (*The Mathematical Gazette*, v. 94, n. 531, Nov. 2010), let's introduce angles α, β, ... as shown:

The focus of our interest are triangles ADE and BCE. In these triangles,

AE = BE, AD = BC, ∠AED = ∠BEC.

We are in a position to exploit the SSA situation. There are just two possibilities: Either the two triangles ADE and BCE are congruent, or

2α + β + γ = 180°,

α + 2β + δ = 180°.

Adding the two and taking into account that

α + β = 60°.

But, by the Exterior Angle Theorem, ε = α + β, and we are done.

Furthermore, again by the Exterior Angle Theorem, ∠ABE + ∠BAE = 60°, implying

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Copyright © 1996-2018 Alexander Bogomolny