MathDB

A plane rectangular grid is given and a “rational point” is defined as a point

(x, y)

where

x

and

y

are both rational numbers. Let

A,B,A',B'

be four distinct rational points. Let

P

be a point such that

\frac{A'B'}{AB}=\frac{B'P}{BP} = \frac{PA'}{PA}.

In other words, the triangles

ABP, A'B'P

are directly or oppositely similar. Prove that

P

is in general a rational point and find the exceptional positions of

A'

and

B'

relative to

A

and

B

such that there exists a

P

that is not a rational point.

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