Are tangent points allowed in ruler and compass constructions?

[Genady]

In classical constructions with ruler and compass, a construction proceeds from point to point where the points are intersections of lines and circles, i.e., where the lines and the circles cross each other. Are the points where the lines and the circles touch each other rather than cross, allowed as well?

More specifically, if I have constructed two circles and I know that they touch at some point, can I proceed with the construction using this point?

[Genady]

Even more specifically... With ruler and compass I could construct the point where the circles touch as a point where the line connecting their centers cross them. However, what I deal with is a construction using compass only, no ruler. I don't know if constructing that point by crossing only some circles is possible.

[studiot]

Well yes the tangent is perpendicular to the line joining the centres.

So it can be constructed as a perpendicular line anywhere along the common centreline.

Two things.

Firstly constructed using a ruler and compass has two variations.

The otiginal strict meaning was that the ruler could only be used as a straight edge, not for linear measurement.

These days many relax that and allow the use of the ruler as a scale device.

 

Secondly you haven't mentioned if you know the diameters of at least one of the circles ?

[Genady]

4 minutes ago, studiot said:

it can be constructed as a perpendicular line anywhere along the common centreline.

Yes, but as I've clarified in the post before yours, I don't have a centerline, or any line, because I don't have a ruler.

 

5 minutes ago, studiot said:

The original strict meaning was that the ruler could only be used as a straight edge, not for linear measurement.

These days many relax that and allow the use of the ruler as a scale device.

If I had ruler, I would use it only as a straightedge, i.e., in the classical meaning.

 

6 minutes ago, studiot said:

you haven't mentioned if you know the diameters of at least one of the circles

The circle is drawn, and its center is known, so it is not a problem to take its radius with compass. Then, the diameter can be constructed by doubling the radius, which is doable with compass only, i.e., no ruler is needed for this.

[studiot]

I'm not being awkward, but what you are attempting is still not clear to me.

Classical geometric construction is not coordinate geometry.

It starts with a blank sheet.

So what do you mean by the centre is known ?

And what do you mean by I don't have a ruler ?

I'm only trying to help after all.

[Genady]

10 minutes ago, studiot said:

I'm not being awkward, but what you are attempting is still not clear to me.

Classical geometric construction is not coordinate geometry.

It starts with a blank sheet.

So what do you mean by the centre is known ?

And what do you mean by I don't have a ruler ?

I'm only trying to help after all.

I should just formulate the original problem, and all will become clear:

Given two points, A and B, construct a point C, using only a compass, such that B is a midpoint between A and C.

[studiot]

Just now, Genady said:

I should just formulate the original problem, and all will become clear:

Given two points, A and B, construct a point C, using only a compass, such that B is a midpoint between A and C.

Ah I see, thank you I will give it some thought.

[Genady]

Good, but this is not the problem that led to the question in the thread. Here is a solution of this problem:

Spoiler

1. Draw a circle of radius AB with the center at A.

2. Draw a circle of radius AB with the center at B.

3. Mark one of their intersections, D.

4. Draw a circle of radius AB with the center at D.

5. Mark its intersection with the circle of step 2, E.

6. Draw a circle of radius AB with the center at E.

7. Its intersection with the circle of step 2 is the point C.

Now, the problem in question is this:

Given points A and B construct point C in the middle between them using only compass.

I've solved this problem, but my solution relies on being able to mark a point where two circles touch rather than cross each other. I don't know if this is legitimate. That was why I've asked.

[Genady]

I've found that the last problem can be solved also by using intersections of circles, i.e., circles crossing rather than touching at points. It can be done by following constructions in the proof of Mohr–Mascheroni theorem - Wikipedia. Although, these constructions are very long and convoluted. 

[Genady]

And another AI fail. A meaningless "solution":

### Steps to Construct Point C (Midpoint of A and B) Using Only a Compass:

 

1. **Draw a circle centered at A with radius AB:**

   - Place the compass at point A and adjust its width to the distance AB.

   - Draw a circle centered at A with radius AB.

 

2. **Draw a circle centered at B with radius AB:**

   - Without changing the compass width, place the compass at point B.

   - Draw a circle centered at B with radius AB.

 

3. **Find the intersection points of the two circles:**

   - The two circles will intersect at two points. Let’s call these points P and Q.

 

4. **Draw a circle centered at P with radius PA:**

   - Place the compass at point P and adjust its width to the distance PA (which is equal to AB).

   - Draw a circle centered at P with radius PA.

 

5. **Draw a circle centered at Q with radius QA:**

   - Similarly, place the compass at point Q and adjust its width to the distance QA (which is also equal to AB).

   - Draw a circle centered at Q with radius QA.

 

6. **Find the intersection points of the new circles:**

   - The circles centered at P and Q will intersect at two points: one of these points is A, and the other is the midpoint C.

 

7. **Identify the midpoint C:**

   - The intersection point that is not A is the midpoint C between A and B.

 

---

[Genady]

Solved. The answer to the OP question is yes, because I've found a simple way to convert a "touching case" into a "crossing case."

[Trurl]

Are you doing drafting by hand? This construction instructions are often confusing but explain something simple. I will give you some constructions that may help.

 

 

You know to find the midpoint of a segment with a compass to set the compass slightly larger than half the distance of the segment and at the beginning and end is the segment strike an arc with the unknown radius above and below the segment.

 

Take the straight edge and draw a line connecting the 2 intersections of the arc. This is the midpoint of the segment.

 

This is described complexly but is easy technique. Showing is easier than writing.

 

 

I need to look this up to verify but if you know the center of the given circle you could take the radius of the target circle, and draw the radius of the target circle from center of the given circle. This new circle and any point on this circle would result in the tangent circle.

 

 

Finding the center of a circle is the same as the center of a segment.

 

I believe this would be one technique in tangent circles. I will look it up but trying it should prove right or wrong.

[Genady]

2 hours ago, Trurl said:

Are you doing drafting by hand?

No. I am doing constructions using only compass.

 

2 hours ago, Trurl said:

Take the straight edge

The problem specifically states NOT to use straightedge.

[Trurl]

8 hours ago, Genady said:

No. I am doing constructions using only compass

Yes, I know. But constructions are common in drafting. They use to teach it before using CAD.

8 hours ago, Genady said:

The problem specifically states NOT to use straightedge

Not for measuring; only to draw a straight line.

 

If you look up how to find the center of a line you can use the same technique to find the center of the circle.

From the center draw a circle of tangent circle’s radius. From any point on this tangent circle draw a circle of the tangent radius.

Now every tangent radius of the last circle will be tangent to the original circle.

Center of ordinal circle will plus unknown radius. Then unknown tangent circle that is subtracted by original circle. And then the tangent radius that by subtracting gives us the original radius distance to form the tangent. All without measuring.

[Genady]

3 hours ago, Trurl said:

Not for measuring; only to draw a straight line.

The problem does not allow to use straightedge to draw a straight line either. 

[Trurl]

10 hours ago, Genady said:

The problem does not allow to use straightedge to draw a straight line either.

In my example you do not need a straightedge. Simply make 4 arcs with the compass from each quadrant. Use an arbitrary radius that is slightly bigger than the radius of the given circle.

This will give you the center. Proceed with constructing the tangent circles radius. Then from the tangent circle draw a construction circle with the original circle’s radius

Draw a circle that encompasses all these circles. (That would be the diameter of the original circle plus the radius of the tangent circle.)

Any radius of the tangent circle’s radius when drawn with the compass from any point on the circumference of the construction circle will be tangent.

 

I need you to try this and see if it works. I checked my drafting book and this case is not in it.

I will test it. It is also hard to describe in words, so if anything in my description doesn’t make sense let me know.

[Genady]

43 minutes ago, Trurl said:

In my example you do not need a straightedge. Simply make 4 arcs with the compass from each quadrant. Use an arbitrary radius that is slightly bigger than the radius of the given circle.

This will give you the center. Proceed with constructing the tangent circles radius. Then from the tangent circle draw a construction circle with the original circle’s radius

Draw a circle that encompasses all these circles. (That would be the diameter of the original circle plus the radius of the tangent circle.)

Any radius of the tangent circle’s radius when drawn with the compass from any point on the circumference of the construction circle will be tangent.

 

I need you to try this and see if it works. I checked my drafting book and this case is not in it.

I will test it. It is also hard to describe in words, so if anything in my description doesn’t make sense let me know.

1. I don't know which problem you are solving.

2. It should be easy to describe a construction in words if the circles and points are named. Let's try.

a. Let's call the given circle, C1.

b. You say to "Use an arbitrary radius that is slightly bigger than the radius of the given circle." I set the compass to some radius like that.

c. "make 4 arcs with the compass from each quadrant." With the radius set in (b), I make circles C2, C3, C4, and C5 with the centers in A2, A3, A4, and A5, respectively. They intersect in several points with C1 and among themselves.

d. "This will give you the center." Where is the center? The center of what?

[Trurl]

3 hours ago, Genady said:

1. I don't know which problem you are solving.

I was solving the entire problem: How to draw infinite circles tangent to C1 knowing only C1 radius as the given.

I don’t believe you gave Studiot or me exact wording of the problem.

I am working on the fact that the radius of C1 is given. If you know the radius you can just use the compass from any point on the circumference to find C1’s center.

3 hours ago, Genady said:

b. You say to "Use an arbitrary radius that is slightly bigger than the radius of the given circle." I set the compass to some radius like that.

Well if the radius of C1 is given you can refer to answer to question 1. If you have to find the center of C1 (That is it is unknown.), you would need a straight edge to connect the horizontal and vertical lines that meet in the center.

Do you know how to use a compass to divide an arbitrary line by striking an arc from both endpoints of the arbitrary line?

Same thing with circle because the straight lines of the diameters of the circle are straight lines revolved around the center.

 

List the exact words of the problem and I will help you solve it.

[Genady]

15 minutes ago, Trurl said:

I was solving the entire problem: How to draw infinite circles tangent to C1 knowing only C1 radius as the given.

I don’t believe you gave Studiot or me exact wording of the problem.

I am working on the fact that the radius of C1 is given. If you know the radius you can just use the compass from any point on the circumference to find C1’s center.

Well if the radius of C1 is given you can refer to answer to question 1. If you have to find the center of C1 (That is it is unknown.), you would need a straight edge to connect the horizontal and vertical lines that meet in the center.

Do you know how to use a compass to divide an arbitrary line by striking an arc from both endpoints of the arbitrary line?

Same thing with circle because the straight lines of the diameters of the circle are straight lines revolved around the center.

 

List the exact words of the problem and I will help you solve it.

I don't know how you got the problem you describe. I've never mentioned anything like that. I have mentioned two other problems earlier in the thread.

Problem A:

On 1/26/2025 at 11:25 AM, Genady said:

Given two points, A and B, construct a point C, using only a compass, such that B is a midpoint between A and C.

Problem B:

On 1/26/2025 at 12:08 PM, Genady said:

Given points A and B construct point C in the middle between them using only compass.

Emphasis is on "compass only". No straightedge, no straight lines, nil, nada, niet!

The question in the thread was NOT how to solve these problems. I've solved both of them. (Without drawing any straight lines.)

My initial solution used a point where two circles constructed on an intermediate step, touch each other. I was not sure, if it is legitimate to rely on a point that was constructed this way.

However, I've found a way how to construct such a point by constructing another circle which crosses the two tangent circles at the point where they touch each other. This shows that using a point where two circles touch is a legitimate shortcut of this a bit longer but definitely legitimate construction. So, the question in the title of the thread is answered.

If you want to play with them, try the Problems A and B yourself. But remember, no straight lines are allowed.

 

[Trurl]

[Genady]

4 minutes ago, Trurl said:

?

[Trurl]

It says “center” ; “tangent” ; c1 radius.

My lettering is bad but isn’t that what you wanted? The tangent?

[Genady]

11 minutes ago, Trurl said:

It says “center” ; “tangent” ; c1 radius.

My lettering is bad but isn’t that what you wanted? The tangent?

No. I have explained what I wanted in the post which is two posts above this one. Here is what I have explained there:

 

[Trurl]

On 1/26/2025 at 10:25 AM, Genady said:

I should just formulate the original problem, and all will become clear:

Given two points, A and B, construct a point C, using only a compass, such that B is a midpoint between A and C.

 

On 1/26/2025 at 5:48 AM, Genady said:

More specifically, if I have constructed two circles and I know that they touch at some point, can I proceed with the construction using this point?

It could be a miscommunication on my part, but I worked on the solution to “the tangent circles.” 
 

I have shown a way to make infinite tangents between 2 circles of unknown radius.

Can you restate the problem? I like geometric constructions. A drawing would help.

[Genady]

10 minutes ago, Trurl said:

Can you restate the problem?

A drawing would help.

Problem A.

Draw two arbitrary points on a sheet of paper. Call them point A and point B. Find the location of point C such that AB = BC. Use only compass.

Here is the drawing: