Normally in geometric construction problems, we use straightedge and compass. Today, we will look at an unusual type of construction problems where we are only allowed to use compass.
There is a special theorem in mathematics called the Mohr-Mascheroni Theorem. This theorem states that any geometric construction which can be done with compass and straightedge can also be done with compass alone. That means that the use of straightedge is actually not necessary.
For a long time, this theorem was known as Mascheroni's theorem after the Italian mathematician Lorenzo Mascheroni. Mascheroni proved this theorem in his book "Geometria del Compasso" which was published in 1797.
However, the first person who discovered the theorem is actually Georg Mohr, a Danish mathematician. In more than 100 year earlier than Mascheroni, Mohr proved this theorem in a book titled "Euclides Danicus" which he published in 1672.
Mohr's book, Euclides Danicus, was forgotten until it was miraculously found in a bookstore in Copenhagen in 1928. And since then, to acknowledge the contribution from the mathematician Mohr, we have a theorem named Mohr-Mascheroni!
Mohr-Mascheroni Theorem. Any point can be constructed by compass and straightedge can be constructed by compass alone.
Let us look at two simple construction problems:
By Mohr-Mascheroni Theorem, it is possible to use compass alone to multiply and divide a line segment. Today, we will solve the first problem and we leave the second problem for our next post.
Multiply a line segment by compass alone
If we construct two equal circles with radius AB and center at A and B, respectively, then they will intersect at two points X and Y. We obtain two equilateral triangles ABX and ABY. If we keep doing like this then we can construct a lattice of equilateral triangles. Thus, we can solve the aforementioned construction problem.
- multiple a line segment, and
- divide a line segment into a number of equal parts.
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| given AB, by compass and straightedge, it is easy to construct AC = 3 AB |
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| given AB, by compass and straightedge, it is easy to divide AB into 5 equals segments |
By Mohr-Mascheroni Theorem, it is possible to use compass alone to multiply and divide a line segment. Today, we will solve the first problem and we leave the second problem for our next post.
Multiply a line segment by compass alone
Problem. Given two points A and B, only using compass, construct a point C on AB so that AC = 3 AB.
If we construct two equal circles with radius AB and center at A and B, respectively, then they will intersect at two points X and Y. We obtain two equilateral triangles ABX and ABY. If we keep doing like this then we can construct a lattice of equilateral triangles. Thus, we can solve the aforementioned construction problem.
Let us stop here for now. Hope to see you again next time.
Homework.
1. Given two points A and B, only using compass, construct the midpoint of AB.
2. Given two points A and B, only using compass, construct two points $M_1$ and $M_2$ on AB, so that $A M_1 = M_1 M_2 = M_2 B$.
3. Given three points A, B, and C, only using compass, construct the center of the circumcircle of the triangle ABC.
