304. The Balance Point · Where medians meet and ratios hold true

Try to break it

Drag A, B, and C to reshape the triangle. The three medians always meet at a single point G — the centroid — and G divides each median in a 2:1 ratio (the vertex-to-G part is twice the G-to-midpoint part). G's coordinates are simply the average ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3).

How you build it

Construct the centroid of a triangle.

• Place point A on the canvas.
• Place point B on the canvas.
• Place point C on the canvas.
• Draw segment AB to form one side of the triangle.
• Draw segment BC to form one side of the triangle.
• Draw segment CA to complete triangle ABC.
• Find the midpoint D of side BC.
• Find the midpoint E of side CA.
• Find the midpoint F of side AB.
• Draw the median AD from vertex A to midpoint D.
• Draw the median BE from vertex B to midpoint E.
• Draw the median CF from vertex C to midpoint F.
• Mark point G where all three medians intersect. This is the centroid.

The proof, step by step

Prove that the centroid divides each median in the ratio 2 : 1.

• Let D be the midpoint of side BC. Using the midpoint formula, D = ((x₂ + x₃)/2, (y₂ + y₃)/2).
• The centroid G divides the median AD in the ratio 2:1. Using the section formula, G = ((2*dx + 1*ax)/3, (2*dy + 1*ay)/3).
• Substitute dx and dy into the expression for G: G = ((2*(x₂+x₃)/2 + x₁)/3, (2*(y₂+y₃)/2 + y₁)/3).
• Simplify the expression: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). This proves the centroid formula.

Worked example

In triangle PQR, the vertices are P(2, 4), Q(6, 8), and R(10, 2). What are the coordinates of the centroid of triangle PQR?

Centroid = ((2+6+10)/3, (4+8+2)/3) = (18/3, 14/3) = (6, 14/3).

• (6, 14/3) — correct
• (8, 14/3)
• (6, 7/3)
• (8, 7/3)