a. Prove that an isometry that fixes three non-collinear points is the identity.
b. Prove that if S and T are two isometries and T(A)=S(A), T(B)=S(B), T(C)=S(C) then T=S.
c. Prove that if two triangles ABC, DEF are congruent, there is a unique isometry mapping one to the other so that A, B, C are mapped to D,E,F respectively.
d. Prove that every isometry of the plane can be expressed as a composition of reflections. The number of reflections required is either two or three
1. Construct a triangle ABC.
2. Construct a point A0, and a segment A0B0 , congruent with AB
3. Construct a triangle with side A0B0,  and congruent with ABC.
4. Construct the perpendicular bisector l of A0. If A=A0 choose any line through A and denote it by l.
5. Denote by rl the reflection about the line l.
6. Set B1=rl (B), and  C1=rl (C).
7. Denote by m the perpendicular bisector of B0B1. If B0=B1, take m as the line through A0 and B0.
8. Prove that A0 is in the line m.
9. Set C2=rm(C1).
1. CASE 1: C2=C0. In this case, prove that T =rmrl rl
2. CASE 2.C2 is different from C0. Consider the line n through A0 and B0. Shot that T=rnrmrl

e. Given a triangle ABC, and a segment A'B' congruent to AB construct all passible triangles A'B'C congruent to ABC.