construct using unmarked straight edge only

[caffeinemachine]

Let points $A$ and $B$ be given on the plane. The mid point of $A$ and $B$, call it $M$, is also given. Mark an arbitrary point $P$ on the plane. Using unmarked straight edge only, construct the line passing through $P$ and parallel to $AB$.

 

[Opalg]

Let points $A$ and $B$ be given on the plane. The mid point of $A$ and $B$, call it $M$, is also given. Mark an arbitrary point $P$ on the plane. Using unmarked straight edge only, construct the line passing through $P$ and parallel to $AB$.

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Draw a random line through $M$, meeting $AP$ at $X$, and $BP$ at $Y$. Let $Z$ be the point of intersection of $BX$ and $AY$. I'll leave you to figure out why $PZ$ is parallel to $AB$.

Hint: What this does is to construct the line connecting $P$ to the harmonic conjugate of $M$ on the line $AB$ (which happens to be the point at infinity).

 

[caffeinemachine]

View attachment 412​

Draw a random line through $M$, meeting $AP$ at $X$, and $BP$ at $Y$. Let $Z$ be the point of intersection of $BX$ and $AY$. I'll leave you to figure out why $PZ$ is parallel to $AB$.

Hint: What this does is to construct the line connecting $P$ to the harmonic conjugate of $M$ on the line $AB$ (which happens to be the point at infinity).

Thank You. I can conclude now using Ceva's theorem.