p,Q,R are points in the sides BC,CA,AB respectively of a triangle ABC such that
BP/PC=CQ/QA=AR/BR;
AP^2+BQ^2+CR^2 is least when P,Q,R bisect the sides

To tackle this problem, we need to delve into the properties of triangles and the concept of ratios along with the idea of minimizing a certain expression involving distances. The statement you provided indicates a relationship between points on the sides of triangle ABC and how their positions affect the sum of squared distances from the vertices of the triangle. Let's break this down step by step.

Understanding the Triangle and Points

Consider triangle ABC with points P, Q, and R located on sides BC, CA, and AB, respectively. The condition given is that the ratios of the segments created by these points are equal: BP/PC = CQ/QA = AR/BR. This means that the points P, Q, and R divide the sides of the triangle in the same proportion.

Setting Up the Ratios

Let’s denote the lengths of the sides as follows:

• Let BC = a
• Let CA = b
• Let AB = c

Assuming the ratio k = BP/PC = CQ/QA = AR/BR, we can express the segments in terms of k:

• BP = ka, PC = (1-k)a
• CQ = kb, QA = (1-k)b
• AR = kc, BR = (1-k)c

Minimizing the Expression

The goal is to minimize the expression AP² + BQ² + CR². To analyze this, we can use the concept of the centroid of the triangle. The centroid is the point where the three medians intersect, and it divides each median in a 2:1 ratio. When P, Q, and R are the midpoints of their respective sides, they effectively represent a balanced position in the triangle.

Calculating Distances

When P, Q, and R are midpoints, we can express the distances from the vertices to these points. For instance:

• AP = 1/2 * BC
• BQ = 1/2 * CA
• CR = 1/2 * AB

Substituting these into our expression gives:

AP² + BQ² + CR² = (1/2 * a)² + (1/2 * b)² + (1/2 * c)² = (1/4)(a² + b² + c²).

Why Midpoints Minimize the Expression

To understand why this configuration minimizes the sum of squared distances, consider the following:

• When P, Q, and R are not at the midpoints, the distances AP, BQ, and CR increase, leading to a larger sum of squares.
• By positioning P, Q, and R at the midpoints, we ensure that each segment is as short as possible while maintaining the ratio condition, thus minimizing the overall distance.

Geometric Interpretation

Geometrically, the midpoints create a balance in the triangle, leading to the least total distance from the vertices to the points on the sides. This is a manifestation of the triangle's symmetry and the properties of Euclidean geometry.

Final Thoughts

In summary, the expression AP² + BQ² + CR² reaches its minimum value when P, Q, and R are the midpoints of the sides of triangle ABC. This result is rooted in the principles of geometry and the nature of distances in a triangle. By ensuring that the points are symmetrically placed, we achieve the optimal configuration that minimizes the sum of the squared distances from the triangle's vertices.