In an equilateral triangle such as ΔABC, there are several special points that coincide due to its symmetrical properties. Here, we are concerned with two points: the centroid (M) and the incenter (N).
Let's discuss each of them briefly:
1. The centroid (M) is the point of concurrency of the medians of a triangle. A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. In an equilateral triangle, all three medians are congruent, and they intersect each other at the centroid. The centroid has an important property: it divides each median into two segments, one twice as long as the other. This means that the distance from the centroid to the vertex of the triangle (in this case, MA) is two-thirds the length of the median.
2. The incenter (N) is the point of concurrency of the angle bisectors of a triangle. An angle bisector is a line that divides an angle into two equal angles. In an equilateral triangle, all three angles are equal (60°), so the angle bisectors are also the internal bisectors of these angles. The three angle bisectors intersect at the incenter, which is equidistant from all sides of the triangle. Since all sides and angles of an equilateral triangle are equal, the incenter is the same distance from each vertex as it is from each side. In other words, the distance from the incenter to any vertex (in this case, NA) equals the radius of the inscribed circle (the incircle).
For an equilateral triangle, the centroid and incenter coincide; they are the same point. Hence, the point M and the point N are the same point, and the distances MA and NA are therefore exactly the same.
So, when comparing the quantities in Column A and Column B, we can conclude:
Column A (MA) = Column B (NA)
This is because in an equilateral triangle, the median and the angle bisector from the same vertex (point A in our case) not only coincide but also equal each other in length since they both start from a vertex and end at the centroid/incenter, which are the same point.
