Three lawnmowers set off from the three vertices of a triangular field moving clockwise around the boundary. When they reach the next vertex they each take a turn inwards (ie veering to the right of the already cut path ahead) by alpha degrees and then, proceeding in a straight line, all three meet up at a point D inside the field. (They might not all get there at the same time.)
Show that if instead they started out going counter-clockwise and veered left by exactly the same angle alpha at the next vertex, then they would also arrive at a common meeting point H (generally different from D).
Puzzle author: David Odell.
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Solution
Same pair of angles at the vertices leads to an upside down version of ceva’s formula in its trigonometric form. So if they meet on the left figure the must meet on the right. Incidentally, These are the two Brocard points of the triangle.
More solutions: https://x.com/Mirangu1/status/1910592080163782780
2 replies on “Three lawnmowers”
A 2nd proof and more. For much more details see:
archive.org/details/episodes-in-nineteenth-and-twentieth-century-euclidean-geometry-ross-honsberger
chapter 10.
Please replace “A’ is the second point” by “C’ is the second point”.