There are 46 chromosomes that come in 23 pairs, one from mother and one from father. You are talking here about chromosome pair 12.
The way matching works is that it cannot determine whether the match is occurring from the maternal chromosome or from the paternal chromosome. It only determines that either the maternal or paternal chromosome is matching at each point along the chromosome pair.
In your case, on that segment on chromosome pair 12 you have:
- Joseph matches Laura
- Joseph matches Charlotte
- Charlotte matches Joseph (good, that verifies)
- Charlotte matches Laura
- Charlotte matches Edna (7.6 cM)
There are two possible reasons why Edna does not match Joseph.
Charlotte matches Joseph on one of her chromosomes (maternal or paternal), and Charlotte matches Edna on the other.
Charlotte and Edna match not because it is a common segment, but because it is a by-chance match where matching base pairs crisscross between the paternal and maternal chromosomes and appear to the matching algorithm to line up. The smaller the matching segment, the more likely this is to happen. Normally you need a 15 cM segment or longer before by-chance matches become improbable. At 7.6 cM, you always need to be suspicious that this might be a by-chance match.
Laura matches both Joseph and Charlotte so they triangulate meaning the 3 of them match each other on the same segment. Triangulation (without a by-chance match in there) is good assurance that they are matching on the same chromosome.
To verify whether Edna is matching on the other chromosome, or whether Edna's match with Charlotte is a by-chance match, you need other people to compare to. You will have to find at least one other person who matches Charlotte on that segment but does not match Joseph and Laura. Then on GEDmatch, you can do a one-to-one compare of that new person with Edna. If the new person matches Edna, then Charlotte, Edna and the new person all match on the same segment and triangulate. They will be matching on Charlotte's other parent's chromosome from the one she is matching with Joseph and Laura.
If the new person does not match Edna, then either Edna, or that new person, or both are a by-chance match with Charlotte and you'll need to find yet another person to compare with.
The key is to realize (1) there are two chromosomes in each pair that a person can match to, and (2) by-chance matches on small segments are possible. Knowing this allows you to logic through the possibilities.