From the shared SummerMath/1ExplorationsInGeometry/Jim'sJourneys, open the file <hyperbolic_plane_scripts.gsp>. You won't be constructing your sketches on this file, but this file's script tools will be available so long as it is open. When you go to use any one of the script tools, choose Show Script View, so that you can see important information about how to use the tool.

1. Open a new sketch, and draw a large circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points in the hyperbolic plane.

b. Construct a (hyperbolic) equilateral triangle with the two given random points as vertices. (Hint: Mimic the Euclidean proof of Proposition 1 using the script tools in hyperbolic_plane_scripts.gsp. For your compass, in a hyperbolic sense, use the tool called hyperbolic circle and for your ruler, use the tool segment or the tool line.)

c. Test your sketch by dragging the two given random points about the disk. (Dragging beyond the disk isn't part of the test, but it might be interesting. (You now have the start of a tiling of the hyperbolic plane; you can complete the job (optional!) by reflecting (Euclidean inversion) this triangle in its sides and successor triangles in the successor sides - an infinite number of times. Appearance will vary quite a bit, depending on where you start.)

d. Measure each of the interior angles of the triangle by using the measure_angle script tool. Try dragging again.

e. Write a brief description of the results of dragging the given vertices about the hyperbolic plane and of measuring the interior angles of the triangle.

2. Open a new sketch or a new page of your previous sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw three random points and construct an angle. To make a given angle, start with one of the three random points to serve as the angle vertex, and the other two as arbitrary points on the angle sides. Use the segment tool to make the angle sides.

b. Construct a (hyperbolic) angle bisector for a given angle. (Hint: Mimic the Euclidean proof of Proposition 9 using the script tools in the file h-plane_constructions.gsp.)

c. Measure the two angles formed by the proposed bisector using the measure-angle script tool.

d. Write a brief description of your results.

3. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points. To make a given segment joining those points, use the segment tool.

b. Construct a (hyperbolic) midpoint for the given line segment. (Hint: Mimic the Euclidean proof of Proposition 10 using the script tools in the file h-plane_constructions.gsp.)

c. Test the proposed midpoint of the triangle by using the reflect script tool.

d. Write a brief description of your results.

4. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points. To make a given segment joining those points, use the segment tool. Construct another point P that is on the segment.

b. Construct a (hyperbolic) line that passes through a point P on the line and is perpendicular to the given line segment. (Hint: Mimic the Euclidean proof of Proposition 11 using the script tools in the file h-plane_constructions.gsp.)

c.Test your construction by measuring the angles formed by the given line segment and your proposed perpendicular.

d. Write a brief description of your results.

5. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw three random points: P, Q, and R. To make a given line, start with Q and R and use the line tool.

b. Construct a (hyperbolic) perpendicular to the given line through the given point P, not on that line. (Hint: Mimic the Euclidean proof of Proposition 12 using the script tools in the file h-plane_constructions.gsp.) If you follow the proof of Proposition 12, you will need to construct a fourth point (the arbitrary point D referred to in Euclid) that is on the "other" side of hyperbolic line from P.

c.Test your construction by measuring the angles formed by the given line and your proposed perpendicular.

d. Write a brief description of your results.

6. Open a new sketch, and construct a circle to be the boundary circle for the Poincare disk. Make sure that the center of the circle is labeled "A" and the sizing point "B."

a. Draw two random points: P and Q. To make a given line, start with Q and R and use the line tool.

b. Now that you've constructed hyperbolic perpendiculars, how about a rectangle for which the hyperbolic line segment PQ is a side?

c. Write a brief description of your results.