34. The Median's Meeting Point · Where three lines converge

Try to break it

Let's pull the triangle apart. See how the medians stretch and shift, but they never lose their promise to meet at G.

How you build it

Construct the three medians of a triangle.

• Place vertex A of the triangle.
• Place vertex B.
• Place vertex C, clearly off the line AB so the triangle has real area.
• Draw side AB (Segment tool: click A, then B).
• Draw side BC (Segment tool: click B, then C).
• Draw side CA (Segment tool: click C, then A).
• Mark point D at the midpoint of side BC — your click snaps onto the exact middle of BC.
• Draw the median from A: Segment tool, click vertex A, then point D. AD is the median from A to the midpoint of BC.
• Mark point E at the midpoint of side CA — your click snaps onto the exact middle of CA.
• Draw the median from B: Segment tool, click vertex B, then point E. BE is the median from B to the midpoint of CA.
• Mark point F at the midpoint of side AB — your click snaps onto the exact middle of AB.
• Draw the median from C: Segment tool, click vertex C, then point F. The three medians AD, BE and CF now cross at one point — the centroid.

The proof, step by step

Prove that the three medians of the triangle meet at a single point.

• A median connects a vertex to the midpoint of the opposite side.
• Every triangle has three medians, one from each vertex.
• All three medians intersect at a single point called the centroid.

Worked example

In a triangle ABC, the medians AD, BE, and CF intersect at the centroid G. If the length of median AD is 18 cm, what is the length of segment AG?

The centroid divides each median in a 2:1 ratio. So AG = (2/3) * AD = (2/3) * 18 = 12 cm.

• 9 cm
• 10 cm
• 12 cm — correct
• 6 cm