reflections on a circular billiard table, 3 times through same point

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reflections on a circular billiard table, 3 times through same point

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Source: Brazilian Mathematical Olympiad 1986 P1

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#1 h Jul 27, 2018, 6:49 AM • 2 Y

Y by Adventure10, Mango247

A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.

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#2 h Jul 27, 2018, 4:32 PM • 5 Y

Y by khan.academy, ThE-dArK-IOrD, kiyoras_2001, Adventure10, Mango247

Nice P1.

Suppose the ball passes through point $P$ three times. Not hard to show that all the chords of the circle that is part of the trajectory of the ball have equal fixed length, called $d$ . Then, one can easily prove that, there exists at most two chords that pass through $P$ and have length $d$ . This means there's a chord that the ball passes twice, so the angle subject to the chord is rational multiple of $2\pi$ , and so the trajectory will pass this chord infinitely many times, done.

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#3 h Jan 10, 2021, 3:34 PM

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Let O be the centre of the given circle and P be the point through which it passes thrice along chords $A_1B_1$ , $A_2B_2$ and $A_3B_3$
Let D, E and F be their midpoints respectively, i.e, the foot of perpendiculars from O.
Clearly, due to the nature of reflection, O must be equidistant from all the chords along which the ball passes.

(As O must lie on the normals at the points of reflection implying that it must lie on the angle bisector of consecutive chords and hence must be equidistant to all of these chords along which the ball moves.)

Hence, DO = EO = FO but $\angle$ PDO = $\angle$ PEO = $\angle$ PFO = $\frac{\pi}{2}$ which implies that D,E and F are all points lying on a circle with diameter PO (let this circle be $\omega$ ).

But there can be at most 2 such points.

(Since any such point a distance R from O must lie on another circle with centre O and radius R and must also lie on $\omega$ which can only intersect each other at at most 2 points (O lies on $\omega$ means that the 2 circles cannot coincide as O is the centre of one.))

Therefore, some 2 of these 3 points must be the same, WLOG let D $\equiv$ E, this must imply that $A_1B_1$ $\equiv$ $A_2B_2$ . Hence, the ball must move along the same chord at least twice meaning it will then reiterate its path continuing in this manner passing along the same chord and hence through P (lying on this chord), infinitely many times and we are done.

This post has been edited 8 times. Last edited by Gypsy_S, Jan 10, 2021, 4:00 PM

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#4 h Feb 12, 2021, 11:51 AM

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Solution

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wasikgcrushedbi

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#5 h Apr 22, 2022, 4:45 PM • 1 Y

Y by Mango247

ThE-dArK-lOrD wrote:

so the angle subject to the chord is rational multiple of $2\pi$ , and so the trajectory will pass this chord infinitely many times, done.

can someone please explain whats meant by this line

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