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I have the automsomal DNA results for myself, 3 full sisters and 1 half-sister. We all share a mother, and we all tested with familytreeDNA. Our results are uploaded to gedmatch.com.

Our parents are all deceased, as are their parents. We have no first-cousins (on our mother's side) and no co-operative first cousins on our father's side.

How best can I triangulate or phase using the data I have, to work with DNA matches to narrow down the lines on which we match?

I'm assuming that somebody who doesn't match my half-sister but matches me and/or some of my full-sisters is probably a match in my fathers side. And somebody who matches me or one of my full-sisters plus my half-sister is probably a match on our mother's side.

Given the geography, my mother and father are unlikely to have been related.

Is there anything else I can deduce?

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As a group of siblings, the set of your DNA results alone do not give very much leverage in narrowing down which ancestral branches can be excluded when trying to find an ancestor responsible for a DNA match with someone else. For the most part, the results from full siblings simply enlarge the set of DNA sequences available to match against others but do not change their nature.

Having half-sibling results does allow ruling out a branch for some matches. If the person matches both your half-sister and one of you full-siblings, then that person (most likely) is related to all of you through your common parent, your mother (just as you said).

However the converse cannot be relied upon. If you have a match with someone who has no match with your half-sister, that does not mean the source of the match has to be from your paternal side. The match could come from your maternal side but just didn't get passed on to your half-sister. Any particular DNA segment of a parent has a 50%-50% chance of being passed to a child, so if the match is a segment from your mother, your half-sister has a large likelihood (50%) of not receiving it. Hence, a match on a smallish DNA segment not showing in your half-sister's results does not allow you to exclude your mother's ancestors from consideration.

That said, if you and the person match on multiple DNA segments from separate chromosomes, and your half-sister has no matches, then the likelihood you connect with the person via your father's side increases. For instance, if you match segments on 3 chromosomes, then for them to have come from your mother yet have none be inherited by your half-sister would require 3 independent "misses" during her ovum formation, which has a probability of 0.5 x 0.5 x 0.5, or 12.5%. In that case there would be an 87.5% chance that your match is via your father's side.

(If you match on multiple segments on a single chromosome, the situation is much more complicated; it depends on the "genetic linkage" between the segments in question.)

One other minor point of leverage in your situation: If you have a significantly-sized match on your X chromosome, that allows you to exclude your paternal grandfather's side (whether or not your half-sister matches). That is just a result of the normal rule of the X never going from father to son.

These considerations are not too likely to help with the search. That means you will probably have to rely on the genealogical documentation trails of your lines and of your match's lines to find the common ancestors. Once you establish connections with multiple DNA matches, you may be able to narrow future DNA matches if those matches happen to match in overlapping DNA segments on which you match your established matching relatives, or, with lesser confidence, the new matches have other DNA matches with your established matching relatives.

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