Geometry: Bisection of an angle and constructing a right angle

Question

When bisecting an angle, some texts suggest that that the arcs drawn from $D$ and $E$ and intersecting at $C$ should be of the same radius as that of the arc drawn from $O$ but it is not necessary. The arcs from $D$ and $E$ should be of any length greater than $DE$ (Fig 1).

Similarly, in order to construct a right angle at a point on a line, there seem to be two methods- drawing a large arc from the point and keeping the same radius drawing four more arcs to finally meet at $U$, then joining the intersection point $U$ and the point on the line (Fig 2).

Another method of constructing the right angle can be drawing a semicircle from the point on the line and then bisect it (Fig 3).

My point is, are there two different ideas working here where in the first we keep the radii of the arcs same and in the second one radii can be of any length (greater than some minimum length)? If we can bisect an angle simply by taking any radius of the arc and also construct a right angle by simple method of bisecting an $180^\circ$ angle then why there is a long process of constructing two $60^\circ$ angles at the Point $P$ (Fig 2) then bisecting the other $60^\circ$ angle to get a $ 90^\circ$ angle?

Answer 1

For the first, there is nothing different going on. Draw $DC$ and $EC$. Triangles $ODC$ and $OEC$ are congruent in either case as corresponding sides are the same, so the angles at $O$ are equal.

For the second, either method works. I have never seen the complicated one, but you are correct it constructs a $60^\circ$ angle and bisects it. They both create the same perpendicular. Whether you think bisecting a segment vs. bisecting an angle means "something different is going on" is in the eye of the beholder.

Answer 2

Probably it is a matter of simpler implementation. If you do not need to change the compass radius, it is good practice not to, as such a change would be an extra step to both present and implement.

Answer 3

We should remember that Euclid's compass was "collapsible": it allows you to trace a circle, but given its center and one point on it. That's why many constructions stick to this paradigm and avoid, when possible, to add extra points, even if one of the first Propositions in the Elements showed the equivalence between collapsible and rigid compass.

And collapsible is also the compass allowed in Euclidea, where using as few "moves" as possible is one of the fundamental goals to achieve. That's another reason of its popularity.