Question: What does an example of a DWEG house look like? It would help to have a hint.
To see two examples, follow this link.
Question: Is there a definition for length of a D-segment?
Yes and no. There is a definition of congruence of segments based on line reflection (of course this means DWEG line reflection in DWEG lines when we are in DWEG geometry). This is what you need to define a circle or a midpoint.
Two segments are congruent if one can be transformed to the other by a D-line reflection (i.e. E-circle inversion or E-line reflection in the D-line, which is an E-circle or an E-line) or a composition of such reflections.
For example, M is the midpoint of AB if M is on D-line AB and AM is congruent to MB. The circle with center A through B is the set of all points P so that AP is congruent to AB. This means you can figure out what is a perpendicular bisector in DWEG.
See more about distance below.
Question: Do the two points defining the segment have to be on a E-circle with the same radius and same coresponding angle to be equal length in DWEG?
No. Take almost any example of a D-line AB and reflect it in another D-line n which is not orthogonal to it. Then the new D-line A'B' will be an E-circle with a different radius but the D-segments AB and A'B' are congruent. Also the angle on the circle taken up by the E-arcs that are the D-segments will almost always be different.
A good example of this is the lab exercise about building a ruler by constructing "equally spaced" points by repeated reflection. The sequence of points you get has congruent segments between successive points. But the Euclidean arc angles become smaller and smaller if you construct a lot of points. This has to be so, since you have to fit an infinite number of congrent D-segments on a D-line which is an E-circle with only 360 degrees of angle.
Question: OK, so I don't need distance to answer a lot of questions. Is there a distance function? I still want to know.
Yes. There is a way to measure distance with a formula. It is a somewhat complicated formula using the idea of cross-ratio which we have avoided so far.
How do we know what the distance function is? The idea is to work backwards from what we did in 444 where we started with distance and defined transformations called isometries that preserve distance. Then we proved that the isometries were all compositions of E-line reflections.
The idea with our approach to the DWEG model is to say in advance what the isometries will be (and thus congruence of segments) and then find a distance so that two segments are congruent if the endpoints are at the same distance.
We have already said what line reflection is in DWEG, so we will take the DWEG transformations to be the products (compositons) of line reflections. Our method of measuring congruence is that two segments are congruent if there is a DWEG tranformation that takes one to the other. Then we can use these segments as measuring sticks to figure out what a distance measure should be that agrees with this idea of congruence. If we pick out one segment as a unit length, we can compare other segments to it by moving them around and comparing them. (There are important details to work out here, and proportion plays a role.)
This is actually similar to the approach that Euclid uses to build up Euclidean geometry. You start with the straightedge (the line) and the compass (the circle with given center through a given point) and the notion of congruence and build up measurement. However, Euclid did not have the explicit idea of transformation. That came from such mathematicians as Felix Klein in the nineteenth century.
Question: If I can't measure length, how can I construct such important objects as the perpendicular bisector of a segment?
The perpendicular bisector of segment AB is the D-line n that reflects A to B. Since the E-circle n reflects A to B, then it reflects D-line AB to itself, so it is orthogonal to this D-line. Also, if M is the intersection of the two D-lines n and D-line AB. Since MA is reflected to MB, then these segments are congruent, and M is the midpoint. Putting this together, if a D-line n reflects A to B, then it is the D-line perpendicular to D-line AB which passes through the midpoint of AB.