How does Y-DNA “triangulation” to a male ancestor 200 years ago work to prove his y-dna and my paternal line?

Proof" of a paternal patriarch's ydna using proper selection of descendant testing could perhaps be defined as a confirming scientific test of a documented paper trail.   Proof in this case would not prove in court who fathered whom, but it would prove the Y-DNA of a distant male. 

 Allow me to be simplistic to make my point.   First, let’s agree that Y-DNA passes largely unchanged from father to son - and only to the son. 

 If your ydna is tested to be the same as the ydna of your documented  (Bible, birth records, deeds, etc) brother then you know the Y-DNA of your biological father because you both inherited his Y-DNA.  If your brother’s YDNA does not match you, then there was a “non-paternity event” and you have no proof of your father’s YDNA.  (Non-paternity events are said to occur 1.3% to 4% per generation.)  Assuming they match, if you also have records which name the father, then you have "proof" that that recorded father’s Y-DNA is the same as yours and your brother.  Without the records, you can only state that the father was a male who is also in the same male line; that is, your father could have been your uncle for example.   So we can establish the Y-DNA of one's father using records and Y-DNA tests of two or more of his sons. 

 If we agree on that point, we can proceed.  

 If a male descendant of grandfather's son A matches the ydna of a male descendant of grandfather's son B AND we have records to show that A and B were sons of grandfather along with records of their descendants, then we can "prove" the Y-DNA of grandfather.  (It is always better to have confirming tests of male descendants of as many sons of grandfather as possible.)

 Continuing with this logic, we can "prove" the Y-DNA of a man living in 1800 if we have good family records and matching Y-DNA tests of male descendants of at least two sons of that 1800 man. 

 Additional information:   

When matching y-dna, it is very beneficial to have more than 12 matching markers, since often we find different surnames matching at 12 markers indicating a common ancestor far, far back in time.  15 Phelps "cousins" were able to prove their Phelps lines of NC & Va back to 3 patriarchs in the 1700s.  However if one is not able to accomplish that difficult research, one may be able to confirm at least part of the way back to a man living in the mid 1800s - and hopefully assume the "tentative" ydna of his male ancestor.  If not that, then the proof should go to back to at least one's great grandfather.   It is not advised to test descendants of a living person because of the unpleasant implications when there is no match.

 Therefore, a single test of a surname line can indicate the "tentative" YDNA for the entire line, but it can be very misleading when proving a paper trail of male ancestries. 

 The first two y-dna tests back to a common ancestor in the 1800s may result in a mismatch.  This means somewhere there has been a non-paternity event  (name change, adoption, etc).  Additional tests must be done on additional descendants to indicate in which line the NPE occurred.

 Below is a copy from DNA Heritage's web page which may help illustrate these points.