Right Triangles on Sides of a Square
What is this about?
A Mathematical Droodle

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.

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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.

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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