Thanks for the question. Based on your description, it sounds like the area you are describing between the bridge and the tip is primarily the middle vault (septum/upper lateral cartilages) of the nose. The middle vault can collapse following surgery if not adequately supported, giving the patient an Inverted-V Deformity. This may be what you are referencing when you describe your "triangle roof" appearance. Of course without seeing and examining you I cannot be sure, but that would be my best guess.
The treatment of an Inverted-V Deformity involves the placement of spreader grafts between the septum and upper lateral cartilages, to lateralize or push out the upper lateral cartilages and thereby widen the middle vault. This can improve your ability to breathe, as collapse of the upper lateral cartilages can narrow the nasal airway. Aesthetically, this should give you relatively straight lines running between the head of the eyebrows and the nasal tip. In male noses in particular, as you had pointed out, this consistent width in the mid-portion of the nose can be desirable from an aesthetic standpoint.
My personal philosophy when it comes to rhinoplasty is that there is no single "ideal male (or female) nose" for everyone. Rather, with thorough discussion and preoperative computer imaging, I work with the patient during the consultation to plan to refine the nose, such that it will blend well with the face and suit the patient's aesthetic as well as my own. Revision rhinoplasty can be challenging, so be sure to do your due diligence in deciding whether to proceed and with which surgeon. I wish you all the best.