Remember Regiomontanus’s rugby problem? Expressed in geometrical terms, the task was this: given two points A and B (the goalposts) and a third point T (the location of the try) lying on the same straight line, find a point P on the line through T perpendicular to ABT, such that the angle is maximised.
The key to solving this geometrically is a lovely little circle theorem. If we draw any circle through two points A and B and pick a third point C on the arc AB, then the angle is the same whatever point C we pick. Conversely, if we are given two points AB and fix an angle , then the locus of all the points C for which is, up to reflection, the arc of a circle passing through A and B. There’s a nice interactive demonstration here, and a sketched proof here.
(You may be familiar with the special case of this theorem in which AB is the diameter of a circle: pick any other point C on the circumference and will be a right angle. This is Thales’ theorem: it’s said that it’s one of the oldest theorems in mathematics, and that Thales was so chuffed when he proved it that he went straight out and sacrificed an ox in gratitude to the gods. I guess most of us feel like that some days.)
Knowing that this theorem is the tool for the job, the rest is almost easy. Given two fixed points A and B, the smaller the circle is the larger the angle will be. (This is “obvious”, and can be proved by taking C to be the point on the circle furthest from the midpoint of AB and considering isosceles triangles.) So to find P in our original problem, we want to find the smallest circle through A and B such that one of the points on the circle also lies on the dashed line through T. But that just means we need the circle through AB which is tangent to the dashed line. We construct this circle (doing so is a limiting case of the famous Problem of Apollonius), and take the kick from the point P at which it touches the line.
Neat, huh? (Mind you, according to the official laws of rugby, the kicker only has one minute to take the kick once s/he has declared an intention to do so… so I guess rugby players must be really good at doing plane geometry problems at speed!)