Circumscribed Circle

    One of the early results of Euclidean geometry that one comes across, or at least did when 'proof' was considered to be a worthwhile pursuit, was how to find the centre of a circle that will pass through the three vertices of a triangle. The perpendicular bisectors of two of the sides of a triangle meet at a point P. An argument using congruent triangles establish this as the required centre of the circle and also shows that the third perpendicular bisector passes through this point.
    Since the result does not make use of the parallel axiom of Euclidean geometry it is also a result in the hyperbolic plane, but some care needs to be taken. The correct statement of the result is as follows.
     
    If the perpendicular bisectors of two sides of a triangle meet at a point then this is the centre of the circumscribed circle.
    The word 'if' is important. In Euclidean geometry any diagram that we care to draw makes it 'obvious' that any two perpendicular bisectors will meet. However this statement does need proof and the proof involves the parallel axiom which alerts us to the fact that it does not necessarily hold in hyperbolic geometry.
     
    In H2 we have three cases for a triangle 
    • the perpendicular bisectors  meet,
    • are parallel,
    • diverge, or are ultra parallel. 
    The following animated gif taken from a Cabri  file demonstrates the three cases.. 

    The perpendicular bisectors of the sides AB and BC do not always meet. If they do then P is the centre of the non Euclidean circle that passes through the three vertices of the triangle. 

    Download

    The Cabri figure for the above can be downloaded.    circum.fig

    A Cabri Java page for this construction.

    If you want add to the construction when you have opened the figure you will find the hyperbol.men commands that were used to draw the figure attached to the Cabri menu for macro construction. These will give you limited capabilities, but in general you will need to open  hyperbol.men file as well.

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