# Right Triangles on Sides of a Square

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe 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.

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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

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

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Copyright © 1996-2018 Alexander Bogomolny