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Alexander Zouev 000051 - 060
Extended Essay – Mathematics Alhazen’s Billiard Problem
Antwerp International School May 2007
Word Count: 3017 -0-
Alexander Zouev 000051 - 060
Abstract The research question of this Mathematics Extended Essay is, “on a circular table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge”. In investigating this question, I first used my own initial approach (which involved measuring various chord lengths), followed by looking at a number of special cases scenarios (for example when both balls are on the diameter, or equidistant from the center) and finally forming a general solution based on coordinate*…show more content…*

“Don Solves the Last Puzzle Left by Ancient Greeks.” Daily Telegraph. April 1, 1997,
Issue 676.
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Alexander Zouev 000051 - 060 border? The law of reflection states that that the angle of reflection and angle of
incidence are equal, with each angle being measured from the normal to the boundary (line indicating the border)7. In figure 1, the incident path θi must have an angle equal to the reflected path θr.
θr
θi
Figure 1) Law of Reflection, θi = θr
The boundary in our case would be a tangent line drawn to the point on the border of the circle where the ball A bounces off the circular side to ball B (Figure 2).
Figure 2)
B
C
A
θr θi
Another way to express this problem is, “to describe in a given circle an isosceles triangle whose legs pass through two given points made inside the circle” 8. This is useful because it allows us to relate the billiard balls to chords within the circle. Observe Figure
Henderson, Tom. “Reflection and Its Importance”. The Physics Classroom. Dated 2004. Viewed 12 March 2005. Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover Publications New York, 1965. 197-200
8 7
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Alexander Zouev 000051 - 060 3: ball A and ball B are located within the circular billiard table with the table’s center at point C. Ball A needs to make contact with the border at point T in order to strike ball B. If we extend the path that ball A must take to the opposite

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