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Prove that there cannot be 5 collinear points

Source: MTRP 2019 Class 11-Short Answer Type Problems: Problem 6 :-

April 9, 2020
number theorySetscoordinate geometry

Problem Statement

Consider a finite set of points, Φ\Phi, in the R2\mathbb{R}^2, such that they follow the following properties :
[*] Φ\Phi doesn't contain the origin {(0,0)}\{(0,0)\} and not all points are collinear. [*] If α∈Φ\alpha \in \Phi, then −α∈Φ-\alpha \in \Phi, cα∉Φc\alpha \notin \Phi for c≠1c\neq 1 or −1-1 [*] If α, β\alpha, \ \beta are in Φ\Phi, then the reflection of β\beta in the line passing through the origin and perpendicular to the line containing origin and α\alpha is in Φ\Phi [*] If α=(a,b), β=(c,d)\alpha = (a,b) , \ \beta = (c,d), (both α, β∈Φ\alpha, \ \beta \in \Phi) then 2(ac+bd)c2+d2∈Z\frac{2(ac+bd)}{c^2+d^2} \in \mathbb{Z}
Prove that there cannot be 5 collinear points in Φ\Phi
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