1) Who is the author of the first comprehensive text on geometry? When and where was
it written? (4 points)
2) State three theorems from the Elements which you find important and which speak
about both lines and circles. (3 points each)
3) Define the following concepts (2 points each):
a) altitudes, b) medians, c) centroid, d) orthocenter, e) incenter, f) circumcenter
4) Define Euler line. What can you say about the position of the three points defining
this line? (4 points)
5) Define the nine point circle. Explain what are the nine points. What can you say
about the nine-point center? (6 points)
6) State Pasch axiom. (4 points)
7) State the circle-circle intersection axiom (E) and the circle-line intersection property
(CLIP). (4 points)
8) State the plane separation theorem and explain how are the sides of a line defined. (4
points)
9) Define an angle and the interior of an angle. (4 points)
10) What is Hilbert’s plane? What is Euclid’s plane? (4 points)
Solve one of the following two problems, each worth 15 points. You may solve
more for extra credit.
Problem 2. a) In a triangle the orthocenter coincides with the circumcenter. Prove that
the triangle is equilateral.
b) In a triangle the nine-point center and the circumcenter coincide. Prove that the
triangle is equilateral.