centre of mass of hollow triangular frame will lie where, inside or outside frame
Answers
Answer:
It will always be inside the triangle.
Explanation:
N.B. This answer was already provided a year ago, and it is being resubmitted after it was deleted (by error?) with the stated reason being "The answer was copied", which it most certainly was not (any check of my history would find only well-thought out, clear and careful responses of my own!). If the deletion was not a mistake, then my guess is that issue was taken with the fact that the original response included a reference (albeit not a link!) to a third party site (and hey, apart from being helpful, referencing is actually professional!). So to be sure, the reference has been left out of this resubmission (be careful not to refer to other web sites, everyone!). The rest of the original response is reproduced though as it is likely to be educational to anyone interested in this question. Enjoy!
Original answer (minus referencing... search the web for key terms if you want to learn more).
It will always be inside the triangle.
In fact, it is the point known as the Spieker Centre, which is the incentre of the medial triangle (there are so many different "centres" of triangles!). As such, the point in question lies inside the medial triangle, which in turn is entirely inside the original triangle.
To see why this point is the centre of mass of the triangular frame, start by showing (this is a nice little exercise) that for a triangle ABC, with sides of lengths a, b, c opposite vertices A, B, C, respectively, the incentre I is the weighted average of the vertices, weighted by the lengths (i.e. masses) of the opposite sides. That is
I = (aA + bB + cC) / (a + b + c).
Then, let D, E, F be the midpoints of BC, CA, AB, respectively. The centre of mass, S, of the frame is then the weighted average of these midpoints, weighted by the sides' lengths. Since each side of the medial triangle is half the corresponding side of ABC, this is the same as the weighted average of D, E, F with weights taken from the sides of DEF, and this, as noted above, is the incentre of triangle DEF.
Thus the centre of mass of the frame of ABC is the incentre of the medial triangle DEF.