arc bisection

[becko]

In geometrical constructions using compass alone, we should restrict ourselves to two fundamental constructions:

1. To find the points of intersection of two circles.
   
2. To draw a circle with given center and radius.
   

 

Can anyone tell me how to find the middlepoint of a given arc using only a compass?

[daniel_haxby]

This might help:

http://www.cut-the-knot.org/do_you_know/compass.shtml

 

The question you ask is not immediately addressed by the constructions given, but the website assures the reader the proof/method you seek is available as a combination of those constructions given. I didn't have the patience to check if this is the case! But it looks like what you're after.

[freelancejak]

Try this:

 

1. Draw an arc of greater than halfway (just estimate) from point A so that it crosses the line AB.

2. Draw an arc of greater than halfway from B so that it crosses the line AB.

3. There should be two points where the arcs intersect, one on each side of AB, connect those two points.

4. The new line should cross AB and be orthogonal to it, where they meet should be the midpoint from A to B.

[becko]

This might help:

http://www.cut-the-knot.org/do_you_know/compass.shtml

 

I saw the website. There's a construction for the middlepoint of the arc (http://www.cut-the-knot.org/do_you_know/compass7.shtml). But this construction uses the compass to transfer distances (when the circles with radius OE and centers at C and D are drawn; we cannot transfer the distance OE and use it as a radius to draw a circle around C or D).

 

I've already convinced myself that by the use solely of a compass that transfers distances, you can accomplish all the constructions allowed by the use of straightedge and compass. My question is whether these constructions can be achieved by a compass that doesn't transfer distances (the classical Euclidean compass that brakes if lifted from the paper).

 

I should make a small change in my previus description of the fundamental constructions allowed by the compass:

 

1. To find the points of intersection of two circles.
   
2. To draw a circle with given center passing through a given point.
   

 

Now it is clear that this compass cannot be used to transfer distances.

 

All I need is a valid construction of the arc bisector. If I have this construction, I can prove that all the constructions allowed by straightedge and compass can be done using the Euclidean compass alone.

[becko]

Try this:

 

1. Draw an arc of greater than halfway (just estimate) from point A so that it crosses the line AB.

2. Draw an arc of greater than halfway from B so that it crosses the line AB.

3. There should be two points where the arcs intersect, one on each side of AB, connect those two points.

4. The new line should cross AB and be orthogonal to it, where they meet should be the midpoint from A to B.

 

This construction uses the straightedge (to draw the line between two points at step 3 and to find the point of intersection of this line with AB at step 4). I need a construction that uses only the Euclidean compass, that doesn't transfer distances.