Just follow the definition!
1 4, it can be proved that the three sides are equal, and the projection of the vertex on the bottom is the center of the triangle on the bottom (because it is a regular triangle, the four centers are one).
2:
A triangular pyramid with an equilateral triangle at the bottom and an isosceles triangle at the side is a regular triangular pyramid.
This can be cited as a counterexample:
Take the regular triangle ABC as the base, make a straight line perpendicular to ABC through point A, and take a point P on the straight line to connect Pb and PC.
PB=PC
Triangle PBC is an isosceles triangle.
But obviously the triangular pyramid is not a regular triangular pyramid.
3. This problem can also be used as a counterexample:
A triangular pyramid P-ABC, the dihedral angle formed by the surface PBC and the bottom ABC is obtuse. At this time, the projection of the point P on the surface ABC falls outside the triangle ABC shape. In addition, the heights of AB and AC passing through point P will be on their extension lines.
But at this time, the condition of equal height on both sides is satisfied!