Download Hyperbolic Constructions using the Poincare Disk Model

Hyperbolic Constructions using the Poincare Disk Model. Maine GeoGebra Conference 2013 Ryan D. Hedstrom, University of New England [email protected] To start: 1. Open GeoGebra on your computer. Go to Help or press F1. 2. In the search window type in “Hyperbolic Geometry in the Poincare Disk” 3. Click on the first link “GeoGebra Newsletter 2012-­‐05”. 4. Scroll down to Tutorials and Materials. And click on the link that says “Hyperbolic Geometry in the Poincare Disk” 5. Go to “download” and download the file. It will appear as “Material-­‐7005.ggb”. Save it to your desktop. 6. Open the file and a new copy of GeoGebra should open. Tasks: (For each one of these, open a new copy of “Matrial-­‐7005.ggb” from your desktop. 1. Construct an equilateral triangle using the circle tool as a “compass”. Create two hyperbolic circles such that the centers of the circles are on the other circles. Draw three segments to make the triangle. Use the hyperbolic measurement to verify that it is indeed an equilateral triangle. 2. Create any triangle using hyperbolic segment. Use the hyperbolic angle tool to measure the angles (the three points must be selected clockwise!). You can add the three measures to verify that the sum of the angles is less than 180. 3. Can you make a Rhombus and verify that it is a Rhombus? 4. Can you make a Square and verify that it is a square? 5. In Hyperbolic Geometry there are two special types of quadrilaterals a. The Lambert Quadrilateral which has 3 right angles. Can you make this? b. The Saccheri Quadrilateral has 2 right angles and two opposite congruent sides where they are not the side between the two right angles. Can you make this? 6. Triangle Centers. Create a triangle to start with three line, not line segments. a. Construct the Medians of the triangle by connecting a segment from the vertex to the midpoint of the opposite side. Create a point at the point of concurrency and label it the centroid. In this model, the centers will disappear as an artifact of the construction so move a vertex around and labe the vertex again so that it always appears. In Euclidean Geometry, the distance from the vertex to the centroid divided by the distance between the centroid and the midpoint is a ratio of 2:1. Does this hold true in the Hyperbolic Model? (Use the command HyperbolicDistance[<point>,<point>]’ in the Input line to verify this. b. In the same construction (hiding lines will help), construct the altitudes of the triangle using the hyperbolic drop perpendicular and label their concurrent point as the orthocenter. Remember to move the verticies around and do this twice. c. In the same construction, (hiding lines again), construct the perpendicular bisectors of the triangle and label their point as the circumcenter. Construct a hyperbolic circle from the circumcenter to a vertex of a triangle. Is the cicumcircle always consistent? (hint: when you move the construction around, the center may disappear. Just construct a new intersection and a new circle so that when one disappears, the other reappears). Does the circumcenter always exist in the hyperbolic model? d. In Euclidean Geometry, the Centroid, Circumcenter and the Orthocenter are collinear. Is this true in the hyperbolic model when all three appear? 7. Can you construct the incircle such that as you move the construction around, the circle is always inscribed in the circle? Remember that the incircle is formed from the intersection of the angle bisectors (incenter). Pay attention as to how you construct the radius and you may have to do it twice as you move the circle around.