# Right Triangles on Sides of a SquareWhat is this about? A Mathematical Droodle

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Explanation

The applet attempts to suggest the following statement:

Four identical right-angled triangles have been added to a square, two on a pair of opposite side on the inside, the other two on the outside. Prove that the vertices housing the right angles are collinear.

### 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.

 What if applet does not run?

There are many ways to establish the truth of the assertion. For example, triangles AMQ and CPN are equal and equally inclined to the parallel sides AD and BC, implying MQ||NP. The same can be said about triangles MBN and PDQ so that MN||PQ. The required collinearity follows.

The applet employs extra construction to make the picture more symmetrical by adding triangles ADS and BCR and forming an internal square NUQV. The two squares PRMS and NUQV have parallel sides and, by symmetry, they share the center. Thus there diagonals coincide.

### References

1. D. Wells, Hidden connections, double meanings: A mathematical exploration, Cambridge University Press, 1988, p. 10

• Equilateral Triangles On Sides of a Parallelogram
• Equilateral Triangles On Sides of a Parallelogram II
• Equilateral Triangles on Sides of a Quadrilateral
• Right Isosceles Triangles on Sides of a Quadrilateral
• Similar Triangles on Sides of a Quadrilateral
• Squares on Sides of a Quadrilateral
• Extra Feature of Van Aubel Configuration
• Further properties of Van Aubel Configuration
• Van Aubel's Theorem for Quadrilaterals and Generalization