Rugby Place-kicking Problem

The place-kicking problem in rugby is a lovely problem that I first came across in a presentation by a well loved doyen of mathematics education in South Africa, namely, Tickey de Jager from Rondebosch Boys High School. The problem also appears in his book (De Jager, 1996: 158-161), as well as in Picciotto (1999: 19-24) where it is contextualised (a little differently) within soccer (football).

fig 8

Rugby Place-kicking Problem
Suppose a try is scored at the point 'Try' shown in the dynamic sketch below, then the kicker can kick to the posts from anywhere along the line perpendicular at that position to the tryline.
Where is the best place to kick from? Think a bit about it before exploring the problem further below.
Explore
1) Explore the problem by dragging the point C below.
2) What do you notice about ∠ACB as you drag point C?
3) Can you find a place where ∠ACB is a maximum?
(Think why it is important to have ∠ACB to have a maximum).
4) Can you locate a precise position for the point C that maximises ∠ACB?
5) Check your answers in 3) and 4) by clicking 'Show circle' button.

Web Sketchpad

Rugby Place-kicking Problem

Challenge
6) Can you explain why (prove that) the optimal position for ∠ACB is located at the point where a circle through A and B (in other words, with AB as chord) touches the perpendicular line to the tryline?
Hint: Click on the 'Link to Locus of Optimal Positions' button and then click the 'Show Hint' button. From the exterior angle of a triangle theorem, what is the relationship between ∠ACB and ∠AC'B? Can you use that to prove the result?

Explore More
7) What is the locus of all the optimal positions? (If theoretically we also consider the other side of the rugby poles and tryline as well).
8) Drag the point 'Try' to explore 7).
9) What curve does it form? Can you explain (prove) your observation?

Some Solutions
10) Compare your solutions to 6) - 9) to those of mine in De Villiers (1999) - see references below.
11) Also compare further with more advanced modelling approaches given by Polster & Ross (2010) and Freitas (2014) in the references below.

Different Related Problem
A slightly different, but directly related problem that is often used by teachers at various levels is that of 'maximising the viewing angle' by an observer when looking at something like the statue of Nelson in Trafalgar Square, the Statue of Liberty or even the Sphinx. Equivalently, we can formulate the same problem about finding the 'largest viewing angle' for a painting hanging on a wall (a problem often dealt with in courses of trigonometry and calculus). For more details, go to the following webpages:
Regiomontanus' angle maximization problem (Wikipedia)
Where to stand to look at statues (by John Barrow)
Maximizing the Viewing Angle of a Painting (Wolfram Demonstrations Project)

References
De Jager, T. (1996). More than Just Maths. Pietermaritzburg: Shuter & Shooter.
De Villiers, M. (1999). Place Kicking Locus in Rugby. Pythagoras, No. 49, pp. 64-67.
Freitas, P. (2014). Optimal Ball Placement in Rugby Conversions. SIAM REVIEW, Vol. 56, No. 4, pp. 673–690.
Picciotto, H. (1999). Geometry Labs, Grades 8-11. Emeryville, CA: Key Curriculum Press, pp. 19-24.
Polster, B. & Ross, M. (2010). Mathematical Rugby. The Mathematical Gazette, Vol. 94, No. 531 (November), pp. 450-463.

External Links
Henri Picciotto's Soccer Angles
Henri Picciotto's Homepage (Various useful items for teachers)
Cyclic quadrilateral (Wikipedia)
Hyperbola (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 20 May 2025.