MATH 6118-090
Non-Euclidean Geometry

Exercise Set #4

1. Suppose x is a root of

.

Is the length x constructible? Explain your answer.

2. Recall that the circumcenter of a triangle is the point at which the perpendicular bisectors of the sides meet.

a. Is it possible for the circumcenter to lie on a vertex of the triangle? If so, under what conditions? If not, why not?

b. Is it possible for the circumcenter to line on a side of the triangle? If so, under what conditions? If not, why not?

c. If the triangle is equilateral, where is the circumcenter?

3. Recall that the incenter of a triangle is the point at which the angle bisectors meet.

a. Is it possible for the incenter to lie on a vertex of the triangle? If so, under what conditions? If not, why not?

b. Is it possible for the incenter to line on a side of the triangle? If so, under what conditions? If not, why not?

c. If the triangle is equilateral, where is the incenter?

4. Recall that the orthocenter of a triangle is the point at which the altitudes meet.

a. Is it possible for the orthocenter to lie on a vertex of the triangle? If so, under what conditions? If not, why not?

b. Is it possible for the orthocenter to line on a side of the triangle? If so, under what conditions? If not, why not?

c. If the triangle is equilateral, where is the orthocenter?

5. Recall that the centroid of a triangle is the point at which the medians meet.

a. Is it possible for the centroid to lie on a vertex of the triangle? If so, under what conditions? If not, why not?

b. Is it possible for the centroid to line on a side of the triangle? If so, under what conditions? If not, why not?

c. If the triangle is equilateral, where is the centroid?

6. (See pg 39, #2) The first known solution to the problem of trisecting an angle is attributed to Hippocrates. Draw a perpendicular from point C on one side of the given angle to a point D on the other side of the angle. Then construct a rectangle . Draw the ray and locate a point E on such that . Hippocrates claims now that . Why is this true? How does this violate the “straightedge and ruler” constraint on the constructions?