Construct using unmarked straight edge only

  • MHB
  • Thread starter caffeinemachine
  • Start date
  • Tags
    Edge
In summary, the conversation discusses constructing a line through a given point $P$ that is parallel to a given line $AB$ using only a straight edge. The key step is to draw a random line through the midpoint $M$ of $AB$ and find the point of intersection $Z$ with the extended lines $BX$ and $AY$. Ceva's theorem can then be used to prove that $PZ$ is parallel to $AB$.
  • #1
caffeinemachine
Gold Member
MHB
816
15
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 $AB$.
 
Mathematics news on Phys.org
  • #2
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 $AB$.
View attachment 415

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).
 

Attachments

  • parallel.png
    parallel.png
    3.8 KB · Views: 41
Last edited:
  • #3
Opalg said:
https://www.physicsforums.com/attachments/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.
 

Related to Construct using unmarked straight edge only

1. What does it mean to construct using unmarked straight edge only?

Constructing using unmarked straight edge only means that you can only use a ruler or straight edge without any markings or measurements to create geometric shapes or lines. This technique requires precise hand-eye coordination and mathematical calculations.

2. Why is constructing using unmarked straight edge only important?

Constructing using unmarked straight edge only is important because it allows for the creation of accurate geometric shapes without relying on pre-made measurements. This technique is often used in mathematics and engineering to ensure precise and consistent results.

3. What are some examples of constructions using unmarked straight edge only?

Some examples of constructions using unmarked straight edge only include drawing a perpendicular line, bisecting an angle, and creating a parallel line. These constructions can also be used to create more complex shapes and polygons.

4. What are the benefits of using unmarked straight edge only in constructions?

Using unmarked straight edge only in constructions allows for a greater understanding of geometric concepts and principles. It also promotes problem-solving and critical thinking skills as it requires precise measurements and calculations.

5. Are there any limitations to constructing using unmarked straight edge only?

While constructing using unmarked straight edge only is a useful technique, it does have its limitations. This method may not be as efficient or practical when creating complex or detailed shapes. It also requires a high level of skill and practice to achieve accurate results.

Similar threads

Replies
4
Views
716
Replies
10
Views
1K
Replies
8
Views
2K
Replies
9
Views
547
  • General Math
Replies
8
Views
1K
Replies
4
Views
996
Replies
10
Views
2K
Replies
36
Views
4K
  • General Math
Replies
6
Views
724
Back
Top