MathDB

Let

Oxy

be a fixed rectangular coordinate system in the plane. Each ordered pair of points

A_1, A_2

from the same plane which are different from O and have coordinates

x_1, y_1

and

x_2, y_2

respectively is associated with real number

f(A_1,A_2)

in such a way that the following conditions are satisfied:
(a) If

OA_1 = OB_1

OA_2 = OB_2

and

A_1A_2 = B_1B_2

then

f(A_1,A_2) = f(B_1,B_2)

.
(b) There exists a polynomial of second degree

F(u,v,w,z)

such that

f(A_1,A_2)=F(x_1,y_1,x_2,y_2)

.
(c) There exists such a number

\phi \in (0,\pi)

that for every two points

A_1, A_2

for which

\angle A_1OA_2 = \phi

is satisfied

f(A_1,A_2) = 0

.
(d) If the points

A_1, A_2

are such that the triangle

OA_1A_2

is equilateral with side

1

then

f(A_1,A_2) = \frac12

.
Prove that

f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}

for each ordered pair of points

A_1, A_2

Problem Statement