**From New Scientist #2184, 1st May 1999**

This is part of a football pitch; C is a corner, CE is a goal-line, CD is a side-line and AB is a side of the penalty area. Rovers have been awarded an indirect free-kick at the point F on AB and the ball is placed at F. Two players, Fay and Patricia, got to G on CD to discuss their plan. Then together they set off running, Fay towards F and Patricia towards P, each at a steady speed. After 10 seconds Fay reaches F and Patricia reaches P. Fay immediately takes the free-kick and kicks the ball along FA, so that it travels at a steady speed. Patricia carries on running at the same speed and in the same straight line. At the moment Patricia reaches AF, the ball reaches Patricia. Our problem is to find the speed of the ball, as follows:

Draw a line which passes through two of the labelled points, A, B, C, … Select a point where your line crosses an existing line and mark it X. Select a labelled point and mark it Y. You are to do this so that the distance between X and Y is the distance the ball travels in 10 seconds.

Which to labelled points should you choose to draw the line through? Which point is Y?

[enigma1028]

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I didn’t write a program to solve this one.

First we extend GP to meet AB at point Z, the point where Pat intercepts the pass from Fay (who is at F).

We can now extend FP to meet (the extension of) CD at point X.

As AB and CD are parallel, the triangles PZF and PGX are similar.

GP is the distance travelled by Pat during the 10s before the pass, and PZ is the distance travelled by Pat during the pass.

During the pass the ball travels the distance FZ, so the corresponding side of the other triangle, XG, will be the same length as the distance that the ball would travel in 10s.

So, the solution to the problem is:

Solution:The line is drawn through F and P. The point labelled Y is G.