The correct answer is:

(3) a point $P$, for which each of the three angles $\angle APB, \angle BPC, \angle CPA$ is 120 degrees.

This point is known as the Fermat point or Torricelli point of the triangle. Here’s why:

Explanation:

1. Fermat’s Principle for Triangle:

   • The point $P$ inside the triangle minimizes the sum $PA + PB + PC$ if and only if the angles formed at $P$, $\angle APB$, $\angle BPC$, and $\angle CPA$, are all $120^\circ$.
   • This is geometrically the optimal point for minimizing the total distance to the three vertices.

2. Conditions:

   • If all angles of the triangle are less than $120^\circ$, the Fermat point lies inside the triangle.
   • If any angle of the triangle is $\geq 120^\circ$, the Fermat point coincides with the vertex of the largest angle.

3. Incorrect Options:

   • (1) The point of intersection of the altitudes (orthocenter) minimizes a different property, related to perpendicular distances, not the total distances $PA + PB + PC$.
   • (2) The centroid (intersection of medians) balances the triangle but does not minimize $PA + PB + PC$.

Tip:

To construct the Fermat point geometrically:

• Construct an equilateral triangle on any one side of the given triangle.
• Connect the vertex of the equilateral triangle to the opposite vertex of the original triangle.
• Repeat this for another side. The intersection of these lines is the Fermat point.

Would you like further explanation, or details on how to prove this?

Here are 5 related questions for you:

1. How can the Fermat point be located geometrically using a compass and straightedge?
2. What happens to the Fermat point if one of the angles of the triangle is $120^\circ$ or greater?
3. How does the centroid differ from the Fermat point in terms of geometric properties?
4. Why is the Fermat point useful in optimization problems involving distances?
5. Can the Fermat point be generalized to polygons with more than three vertices?

Tip:

Always verify the triangle's angles before assuming the location of the Fermat point—it changes if any angle is $\geq 120^\circ$.