In honour of the Six Nations rugby tournament currently taking place, here’s a problem in plane geometry loosely inspired by one of the many dilemmas facing a rugby player.
For those unfortunate enough not to be familiar with the sport: a team can score by touching the ball down beyond the opposing team’s goal-line, scoring five points for a try. After a try is scored, the scoring team has the opportunity to gain an extra two points by converting the try: that is, kicking the ball over the crossbar between the goalposts. This conversion kick must be taken from a point somewhere on the line perpendicular to the goal-line and passing through the point where the try was scored (see the diagram below). The question is: from what point on this line should the kick be taken to make it easiest to score?
We’ll assume that T does not lie between A and B, or the decision becomes easy. We’ll also assume that pure distance is not the controlling factor: what matters is how wide the goal appears to the kicker to be — in more technical terms, the angle subtended by the posts, such as or above. Clearly, if the kick is taken from right on the tryline then and there is no hope of scoring. From the far end of the pitch, on the other hand, the goal looks tiny and there is again no hope. The optimal point must be somewhere in the middle… but where?
This problem can be tackled using a combination of trigonometry and calculus, but it turns out that there’s a very elegant geometric solution, using one of the classic elementary theorems of plane geometry. Can you find it?
PS. According to Miodrag S. Petković’s book Famous Puzzles of Great Mathematicians, this problem goes back to the mathematician and astronomer Regiomontanus in the fifteenth century, although for some reason he didn’t phrase it in terms of rugby. (Being Bavarian, he was probably more into football.)