Trigonometry/Converting One Triangle into Another
- Given two triangles of the same area, can we cut one up into a finite number of pieces which can then be rearranged to make the other?
If we can cut a triangle up and rearrange the pieces to make a rectangle, then we're doing well. Rectangles are easy to work with. If we can get a number of small rectangles, somehow make them into rectangles with one side of length one, then stack them all up on that side into one rectangular bar with one side of one, then we've won. Why? Because the process of cutting and rearranging is reversible. To spell that out:
- We can cut and rearrange the first triangle to make a rectangle with the same area and one side of length one. We can do the same for the other. If we superimpose the cut-lines of the two ways of dividing up this unit rectangle we have pieces which can be rearranged to make either the first triangle or the second.
Well, we can cut any triangle into two right triangles, and each of those right triangles can be cut and reassembled to make a rectangle. So we can certainly get two rectangles out of each triangle. We can cut these up into lots of small rectangles.
Now, how do we convert those rectangles to rectangles with one unit side?
A little experimentation shows that the way to go is via parallelograms. We can make cuts at an angle to get the length of side that we want. And we can cut parallelograms up and reassemble them to make rectangles, whilst preserving the length of one side. Once we've hit on the idea of using parallelograms, and looked at ways of slicing and reassembling them, we see that we don't really need to do anything special with rectangles. Here's how we do it.
We don't have to do our dissection into small pieces at the start. We can do it as we go. Then whatever solution we end up with we can translate into a solution where we did dissect the triangles at the start.
- Back to Hilbert's Third Problem