I have been watching this thread since the beginning, and having known the answer upon reading the question, I was surprised that no one answered when you clarified your definition of a corner, you all but gave the answer in your drawing.
My issue with this brain teaser is not totally with the answer, but also with the words chosen to ask the question.
the way you used the term conjoined is not exactly what it means. coinjoined implies 2 distinct things united or touching.
if you were to remove the triangle from the figure, there is no 'square'.
I am not sure there is a simple way to pose the question, but if you could substitute another word with the meaning you desire the question would be more sound.
having said this, if you do crossword puzzles, liberties like this are taken all the time.
I look forward to more puzzles and thank you for this great thead.