a. Prove that an isometry that fixes three non-collinear points is the
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
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
e. Given a triangle ABC, and a segment A'B' congruent to AB construct all
passible triangles A'B'C congruent to ABC.
- Construct a triangle ABC.
- Construct a point A0, and a segment A0B0
, congruent with AB
- Construct a triangle with side A0B0, and
congruent with ABC.
- Construct the perpendicular bisector l of A0. If A=A0
choose any line through A and denote it by l.
- Denote by rl the reflection about the line l.
- Set B1=rl (B), and C1=rl
- Denote by m the perpendicular bisector of B0B1.
If B0=B1, take m as the line through A0
- Prove that A0 is in the line m.
- Set C2=rm(C1).
- CASE 1: C2=C0. In this case, prove that T =rmrl
- CASE 2.C2 is different from C0. Consider the
line n through A0 and B0. Shot that T=rnrmrl