I was wondering whether the following questions are difficult to solve :
Consider a triangle ABC (defined in euclidean geometry).
Let M be inside the triangle ABC such that the triangles AMB, AMC and BMC have same perimeter. What can be said about existence and uniqueness of M ? If unique, does it correspond to a particular point of the triangle ?
Let N be inside the triangle ABC such that the triangles ANB, ANC and BNC have same area. What can be said about existence and uniqueness of N ? If unique, does it correspond to a particular point of the triangle ?
I tried to obtain some equations starting from points A(0,0), B(1,0) and C(x,y), but I feel this is not the correct way to handle this, maybe I am lacking some point of view involving only elementary tools (say knowledge related to angle/perpendicular bisector and so on). Any comment on these questions?
$M$ is the Veldkamp's isoperimetric point while $N$ is just the centroid.