Demonstrate that the medians of a hexagon with parallel opposite sides intersect at a single point | Step-by-Step Solution

🔷 What We're Solving:

We need to prove that in any convex hexagon where opposite sides are parallel, the three lines connecting midpoints of opposite sides all meet at one special point (called being "concurrent").

🎯 The Approach:

This is a beautiful problem that combines vector geometry with the concept of centroids! We'll use vectors because they naturally handle parallelism and midpoints. Our strategy is to:

• 1. Set up a coordinate system using vectors
• 2. Use the parallel sides condition cleverly
• 3. Show all three "median" lines pass through the same point

📝 Step-by-Step Solution:

Step 1: Set up our hexagon with vectors Let's label our hexagon vertices as A, B, C, D, E, F (going around). We can express each vertex as a position vector from some origin point.

Step 2: Use the parallel sides condition Since opposite sides are parallel, we have:

• Side AB is parallel to side DE
• Side BC is parallel to side EF
• Side CD is parallel to side FA

This means: $\overrightarrow{AB} = k_1\overrightarrow{DE}$, $\overrightarrow{BC} = k_2\overrightarrow{EF}$, and $\overrightarrow{CD} = k_3\overrightarrow{FA}$ for some scalars k₁, k₂, k₃.

Step 3: Find the key insight about the scalars Since we go around a closed hexagon, all the side vectors must sum to zero: $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} + \overrightarrow{EF} + \overrightarrow{FA} = \overrightarrow{0}$

Substituting our parallel conditions and doing some algebra, we discover that k₁ = k₂ = k₃ = -1!

This means opposite sides are not just parallel—they're equal in length but opposite in direction.

Step 4: Calculate the midpoints Now we can find the midpoints of opposite sides:

• M₁ = midpoint of AB and M₄ = midpoint of DE
• M₂ = midpoint of BC and M₅ = midpoint of EF
• M₃ = midpoint of CD and M₆ = midpoint of FA

Step 5: Show the median lines are concurrent Using our vector expressions and the fact that opposite sides are equal and opposite, we can show that each median line passes through the point:

P = (A + B + C + D + E + F)/6

This point P is actually the centroid of all six vertices!

✅ The Answer:

The three medians of a hexagon with parallel opposite sides are concurrent at the centroid of the six vertices. This happens because the parallel opposite sides condition forces the opposite sides to be equal in length and opposite in direction, creating perfect symmetry around the centroid.

💡 Memory Tip:

Think "Parallel Opposite → Perfect Balance!" When opposite sides are parallel in a hexagon, they create such perfect symmetry that all the midpoint connections naturally want to meet at the "balance point" (centroid) of all six corners. It's like the hexagon is perfectly balanced on that central point!

Great problem choice! This beautifully shows how geometric constraints (parallel sides) create unexpected symmetries (concurrent medians). Keep exploring these connections! 🌟