Note that the nine-point centre lies halfway between the circumcentre and the orthocentre. If both of them lie inside the triangle, this cannot happen. Hence, at least one of them has to lie outside. This means that the triangle has to be obtuse (a right-angled triangle does not work because in that case, the orthocentre coincides with a vertex). Hence, the image you should have in mind is that of an obtuse triangle with the obtuse angle being A.
- This reply was modified 1 year, 3 months ago by Sankhadip Chakraborty.