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construct using unmarked straight edge only

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
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 $AB$.
parallel.png

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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caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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.