A pyramid seems to fit the description.
The vertex is a point of polygon or polyhedra that is formed by 2 lines or 3 edges on a polyhedron.
A line that joins any vertex of a polygon (or polyhedron) to any vertex other than itself or adjacent ones is a diagonal.
pyramid
A diagonal is normally defined as a straight line joining a vertex of a polygon with any vertex other than an adjoining vertex (lines joining a vertex to an adjoining vertex would simply be a side of the polygon). Since a triangle has only got adjoining vertices, it has no diagonals. Since there are no diagonals, they cannot bisect one another.
A vertex
A pyramid fits the given description
The vertex is a point of polygon or polyhedra that is formed by 2 lines or 3 edges on a polyhedron.
A pyramid has a polygonal base with all other sides being triangles. As the Egyptians famously demonstrated, a common pyramid has a square base with four triangles meeting at a vertex, but any polygon - regular or irregular - can be used as the base of such a polyhedron.
A polyhedron of which one face is a polygon of any number of sides, and the other faces are triangles with a common vertex.
A line that joins any vertex of a polygon (or polyhedron) to any vertex other than itself or adjacent ones is a diagonal.
pyramid
pyramid An pyramid is a geometric solid of the shape made famous by the royal tombs of ancient Egypt. It is a solid whose base is a polygon and whose lateral faces are triangles with a common vertex
A triangle has fewer than 4 vertices.
The definition of vertices is the plural vertex. A vertex is point of intersection of two or more lines. A vertex can also be used to describe the angular point of a polygon, polyhedron, or other shape.
a square a rectange a triangle a hexagon an octogon a polygon
A vertex
A diagonal is normally defined as a straight line joining a vertex of a polygon with any vertex other than an adjoining vertex (lines joining a vertex to an adjoining vertex would simply be a side of the polygon). Since a triangle has only got adjoining vertices, it has no diagonals. Since there are no diagonals, they cannot bisect one another.