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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?

Curious
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1 Answers1

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$M$ is the Veldkamp's isoperimetric point while $N$ is just the centroid.

Jack D'Aurizio
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    Thanks for this answer! Actually I spent more time on the first part (perimeter). The condition related to the existence is not so easy, I am quite happy I wasn't able to found it... – Curious May 16 '15 at 09:47