December 2001
Ellen Maycock
How do we construct the inscribed and circumscribed circles of a triangle in various geometries? Do similar constructions hold in Euclidean, spherical and hyperbolic geometry? In this lab, we will investigate these constructions and see some of the special features of these circles.
This section should be done as homework before the laboratory period.
Remember that the circumscribed circle of a triangle is the circle that goes through the 3 vertices of the triangle. The center of this circumscribed circle is called the circumcenter. The inscribed circle of a triangle is the circle that touches each side of the triangle exactly once. The center of this circle is called the incenter.
Let's begin with a compass and straightedge construction in Euclidean geometry. You may use pencil and paper or a dynamic software such as Geometer's Sketchpad to do your constructions.
Draw an arbitrary triangle
Construct the angle bisectors of each angle of the triangle. If you are using a dynamic program, convince yourself that the angle bisectors always meet in one point by moving the vertices around. Which circle would have this point as a center? Why?
Construct the perpendicular bisectors of each side of the triangle. As before, convince yourself that they always meet in one point. Which circle would have this point as a center? Why?
You will work through this section with your partner during the laboratory period.
You will use the Lénárt Sphere to consider these constructions in spherical (elliptic) geometry and the dynamic program PoincaréDraw to investigate them in hyperbolic geometry.
You will need to answer the following questions by working on the Lénárt Sphere.
Draw a triangle on the Lénárt Sphere. Then construct the polar triangle associated to triangle . (See below if you are not familiar with the polar triangle).
Draw the angle bisectors of the original triangle . Do they intersect in a point? More than one point? No points? Be sure to consider some extreme cases, such as when all the vertices lie on one great circle. Do the angle bisectors of triangle relate to triangle ? How?
For the cases when you found a common point of intersection, use this point as the center to draw a special circle for the triangle . Is this the same construction you used in Euclidean geometry? Also use this point to draw a special circle for triangle
Now draw the perpendicular bisectors for the sides of triangle . Do they intersect in a point? More than one point? Be sure to consider some extreme cases. For the cases when you found a common point of intersection, use the point to construct a special circle for triangle . Is this the same construction as in Euclidean geometry? Do the perpendicular bisectors of the sides of triangle relate to the triangle ?
Write a conjecture based on your work in this section. Explain in your own words why you think your conjecture is true.
You will need to answer the following questions by working in the computer lab, using the program PoincaréDraw.
Plot 3 points on the screen, and construct the triangle Then construct the angle bisectors of each angle. Do they intersect in one point? Move the triangle around the screen to consider different possibilities, making sure you include some extreme cases.
Try to construct an inscribed circle based on your work in the problem above. Does the construction agree with the construction in Euclidean geometry? Does the construction always work? Why or why not?
Now construct the perpendicular bisectors of each side of triangle . Do they intersect in a point? If so, draw a circumscribed circle for triangle using this point as a center. Use the dynamic property of the program to see if the construction holds for all triangles.
What problems did you run into trying to answer question 11? Can you make a conjecture based on the data? What possibilities can occur with the perpendicular bisectors? When will you be able to construct a circumscribed circle?
This section should be done as homework after the laboratory period.
Write a careful proof that the perpendicular bisectors of the sides of a triangle intersect in a point that is the center of the circumscribed circle in Euclidean geometry.
Write a careful proof that the angle bisectors of a triangle intersect in a point that is the center of the inscribed circle in Euclidean geometry.