Two Symmetric Triangles Are Directly Equidecomposable

The problem #9 in a delightful collection Which Way Did the Bicycle Go? reads:

By a printing mischief, my original copy of the book lacked about 50 pages. It so happened, that the solution to the problem appeared on the missing ones. MAA has eventually corrected the evil. Meanwhile, the problem caused some activity and agitation on these pages.

Curiously, I began by misreading the problem: I got an impression that the problem is to find two pieces that, without flipping, combine in two different ways to create to symmetric triangles. But this would require a single cut. While this can be done for a right triangle, in general a single cut would not make it. Discouraged, I found an incomplete solution: the best I could do in general was to cut an acute triangle in 3 pieces (with 3 cuts) and any triangle into 4 (also with 3 cuts.) The latter is based on the dissection of a right triangle with a single cut.

You can see my solutions below. Nathan Bowler, put things in order with a remark that 2 out of 4 pieces combine into a bigger one that can be moved rigidly, thus solving the problem as stated. In passing, there is another decomposition of the triangle into 3, now quadrilateral, pieces, with 3 cuts.

Upon receiving a viable copy of Which Way Did the Bicycle Go? I have also added the intended solution.

In decomposition by dissection the rearranged pieces may also change their orientation. When orientation is preserved, the two shapes are said to be directly equidecomposable. This is what the problem is about: two equal triangles are directly decomposable even if their orientations differ.

References

1. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996, #9

Equidecomposition by Dissection

1. Carpet With a Hole
2. Equidecomposition of a Rectangle and a Square
3. Equidecomposition of Two Parallelograms
4. Equidecomposition of Two Rectangles
5. Equidecomposition of a Triangle and a Rectangle
6. Equidecomposition of a Triangle and a Rectangle II
7. Perigal's Proof of the Pythagorean Theorem
8. Two Symmetric Triangles Are Directly Equidecomposable
9. Wallace-Bolyai-Gerwien Theorem

Copyright © 1996-2008 Alexander Bogomolny

Solution

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The applet does not specifically show the 2-pieces solution for a right triangle. It can be surmised from the general 4-pieces solution. In the applet, you can drag the vertices of the triangle and the triangle itself. The mirror image of the triangle is created automatically. When satisfied with its shape and position, press the "Move them" button to see how the decomposition works.

Copyright © 1996-2008 Alexander Bogomolny