MathDB

Let

f: \mathbb{R}^2 \to \mathbb{R}^2

be a function having the following property: For any two points

A

and

B

\mathbb{R}^2

, the distance between

A

and

B

is the same as the distance between the points

f(A)

and

f(B)

.
Denote the unique straight line passing through

A

and

B

l(A,B)

(a) Suppose that

C,D

are two fixed points in

\mathbb{R}^2

. If

X

is a point on the line

l(C,D)

, then show that

f(X)

is a point on the line

l(f(C),f(D))

.
(b) Consider two more point

E

and

F

\mathbb{R}^2

and suppose that

l(E,F)

intersects

l(C,D)

at an angle

\alpha

. Show that

l(f(C),f(D))

intersects

l(f(E),f(F))

at an angle

\alpha

. What happens if the two lines

l(C,D)

and

l(E,F)

do not intersect? Justify your answer.

Problem Statement