This type of "triangle" would be the result of calculus-type calculation where one side of the triangle approaches zero in length. The figure will only be a true triangle (by definition, 3 sides, 3 angles) so long as the side is still approaching zero, no matter how small. Once the side becomes zero, the figure will fit the description in the question above, but will no longer be considered a triangle, for practical purposes.
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vertex
the vertex is the common endpoint
Vertex
Every side of a polygon shares endpoints, one each with the two sides next to it.
The sides of a triangle do not meet in a point, so there is no "the common point".