Vol. 5, No. 3, 2012

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Alhazen's hyperbolic billiard problem

Nathan Poirier and Michael McDaniel

Vol. 5 (2012), No. 3, 273–282

Given two points inside a circle in the hyperbolic plane, we study the problem of finding an isosceles triangle inscribed in the circle so that the two points belong to distinct congruent sides. By means of a reduction to the corresponding result in Euclidean geometry, we prove that this problem cannot generally be solved with hyperbolic ruler and compass.

hyperbolic geometry, Alhazen
Mathematical Subject Classification 2010
Primary: 51M04, 51M10, 51M15
Secondary: 51M09
Received: 18 January 2011
Revised: 24 June 2011
Accepted: 26 June 2011
Published: 14 April 2013

Communicated by Joseph Gallian
Nathan Poirier
Department of Mathematics
Aquinas College
1607 Robinson Road SE
Grand Rapids 49506
United States
Michael McDaniel
Department of Mathematics
1903 W. Michigan Ave.
Western Michigan University
Kalamazoo, MI 49008-5248
United States