Course:Introduction to Geometrical Thought and Methods
Midterm Examination, Spring 2009
Total points:110
Note A,B,C...denote points and m n l denote lines.The "segment AB" means
{C/A*C*B}union{A,B} and "ray AB" means the union of segment AB and {C/A*B*C}
∠BAC means the angle given by ray AB and ray AC.
1(20pts)State the first three groups of Hilbert's axioms for Foundation of
Geometry,namely axioms for Incidence(3 axioms),Betweeness(or order)(4 axioms)
and Congruence (6 axioms).
2(20pts) Suppose that A*B*C and A*C*D. Prove that B*C*D.
3(35pts)
a(15pts) Let A,B屬於l and C,D lie on the same side of l.Prove,by first
definig "betweeness" for rays,that either ray AD lies between ray AC and ray
AB , or ray AC lies between ray AD and ray AB
b(20pts)Let ∠BAC=∠B'A'C' and D' lie interior of ∠B'A'C'.Suppose D is a
point such that D,B lie on the same side of line AC,and that ∠DAC=∠D'A'C'
Prove,assuming the result a),that Dlies interior of ∠BAC
4(35pts) The following concerns the arithmetic of segments (here the ordinary
Parallel Axiom is assumed). a)(10pts) Given segments a,b, define the
multiplication a*b. Also state the Pascal's theorem without giving proof.
b(10pts) Prove the distributive law a(b+c)=ab+ac.
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