Using compass and straightedge, we can construct the midpoint of a line segment, and we can easily divide a line segment into, say three equal parts. The question is, is it possible to do these constructions

**with just the compass**.

The answer is "yes"! Indeed, it is possible to construct the midpoint of a line segment, and it is possible divide a line segment into a number of equal parts by using the

**compass alone**. How amazing is that?! Today, we will look at these constructions!

Basic compass-and-straightedge constructions

First, let us recall the basic compass-and-straightedge constructions. In our previous post, we learn about these constructions. These constructions are assumed as obvious and can be employed in more complex construction problems:

- Construction of the perpendicular bisector of a line segment and the midpoint

- Through a point, construct a line perpendicular to a given line

- Through a point, construct a line parallel to a given line

- Construction of the angle bisector

- Construction of an equal angle to a given angle

- Through a point, construct a tangent line to a circle

You can read more about these basic compass-and-straightedge constructions here.

A simple compass-and-straightedge construction problem is how to divide a line segment into a number of equal parts. For instance, suppose we are given a line segment $AB$, using compass and straightedge, how can we divide $AB$ into three equal parts?

We now move on to consider constructions that use compass but not the straightedge.

Construction by compass alone

By Mohr-Mascheroni Theorem, it is then possible to use compass alone to multiply and divide a line segment.Mohr-Mascheroni Theorem.Any point can be constructed by compass and straightedge can be constructed by compass alone.

**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 $$AC = AB \times 3.$$

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 multiply a line segment easily.

**Divide a line segment by compass alone**

Problem.Given two points A and B, only using compass, construct a point D on AB so that $$AD = AB / 3.$$

We can see that the two points C and D are kind of inverse of one another. On one hand, we have $$AC = AB \times 3$$ and on the other hand, we have $$AD = AB / 3$$

This is called the

*inversion operation*. We will learn more about this inversion operation in the future.
Because of this inversion, we have $$AC \times AD = AB^2$$

In geometry, inversion equation like the above is often derived from similar triangles.

If P is an arbitrary point such that AP = AB then the two triangles ADP and APC are similar triangles.
This is because the two triangles share the same angle A and they have equal side ratios $$\frac{AD}{AP} = \frac{AP}{AC}.$$

Since P is quite arbitrary, we will choose a convenient position for P. We will choose P so that the two triangles ADP and APC become isosceles triangles. That is $$PA = PD, ~~~~~~ CP = CA$$

Since $AP = AB$ and $CP = CA$, the point P is an intersection point of the circle centered at A with radius AB and the circle centered at C with radius CA. It means that we can construct this point P by compass alone.

Furthermore, since PD = PA = AB, the point D lies on the circle centered at P with radius equal to AB. By symmetry, we can construct the point D. The construction is as follows:

- Construct the circle centered at A with radius AB and the circle centered at C with radius CA. The two circles intersect at P and Q.

- Construct the circle centered at P with radius equal to AB and the circle centered at Q with radius equal to AB. The two circles intersect at A and D.

So by using compass alone we indeed can construct the point D to divide the line segment into three equal parts.
Let us stop here for now. Hope to see you again next time.

*Homework.*

1. Given three points A, B, and C,

*only using compass*, construct the circumcircle of the triangle ABC.
2. Given three points A, B, and C,

*only using compass*, construct the incircle of the triangle ABC.
3. Go to google.com and search about the geometric inversion operation.

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