MHB Construct using unmarked straight edge only

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To construct a line through point P that is parallel to line AB using only an unmarked straight edge, first identify the midpoint M of AB. Draw a random line through M that intersects line AP at point X and line BP at point Y. The intersection of lines BX and AY creates point Z, which establishes that line PZ is parallel to AB. This construction effectively connects point P to the harmonic conjugate of M on line AB, which is the point at infinity. The discussion concludes with the application of Ceva's theorem to finalize the construction.
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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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caffeinemachine said:
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).
 

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Opalg said:
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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).
Thank You. I can conclude now using Ceva's theorem.
 
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