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

- Construct a triangle ABC.
- Construct a point A
_{0}, and a segment A_{0}B_{0}, congruent with AB - Construct a triangle with side A
_{0}B_{0}, and congruent with ABC. - Construct the perpendicular bisector l of A
_{0}. If A=A_{0}choose any line through A and denote it by l. - Denote by r
_{l}the reflection about the line l. - Set B
_{1}=r_{l}(B), and C_{1}=r_{l}(C). - Denote by m the perpendicular bisector of B
_{0}B_{1}. If B_{0}=B_{1}, take m as the line through A_{0}and B_{0.} - Prove that A
_{0}is in the line m. - Set C
_{2}=r_{m}(C_{1}). - CASE 1: C
_{2}=C_{0}. In this case, prove that T =r_{m}r_{l}r_{l} - CASE 2.C
_{2}is different from C_{0}. Consider the line n through A_{0}and B_{0}. Shot that T=r_{n}r_{m}r_{l}