Note 180

A difficulty is encountered by the fact that the extension of the subject term (S) is equal to that of the proprium term (P), entailing that there is no space for a middle term, implying that we could not prove the conclusion  ( S )  is  ( P ).  Nevertheless, the argument (first syllogism) which leads to :

A triangle ((S))  IS  a figure with the three internal angles (taken together) equal to 180 degrees ((P)).

or, succinctly,  ( S )  is  ( P ),  is correct :
The conclusion is based on the legitimate possiblity to replace the definition of the proprium (DP) by the proprium itself (P). So it is not a demonstration from the more general to the more special (such as :  all mammals are vertebrates, a dog is a mammal, ergo a dog is a vertebrate), but a demonstration based on equivalence.
It is, however, not clear why  "closed figure consisting of three straight lines"  should signify the Essence (that is, the Essence of a triangle), while  "a figure closed by straight lines (not yet given how many such lines), of which (figure) the sum of the internal angles is equal to 180 degrees"  should signify a  proprium. Why couldn't it be the other way around? Here we see the difficulty to distinguish a (specific) proprium from the Essence.
Departing from the human example, in which the nature of man is already stipulated, one distinguishes between an essential formal content (quale) and a formal content that could -- in the act of recognizing things -- be a substitute for the Essence (such a formal content being a proprium). The one is essential, while the other, although coexistent with that being, apparently is not essential. The particular examples may be wrong in some cases, but this does not exclude that such a difference truly exists.
As the reader has seen, we have proposed a solution to this difficulty.

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