Medians and the Centroid: Practice

The centroid division theorem applies to a median. Without the midpoint condition on D, AD need not be a median, so the ratio on that arbitrary segment does not identify the centroid.

Let M be the midpoint of BC. The median AM contains the centroid G. The two distinct points A and G determine one line, so AM is line AG. Its unique crossing with BC is M, hence D is M.

D is strictly between B and C, so B and C are on opposite sides of AD. The interior point E on AC is on the C side. Thus BE crosses AD inside the triangle.

With three collinear vertices, the usual three-sided region is lost and median lines can coincide. There need not be one distinguished intersection point. Right, obtuse, and scalene triangles are allowed when their vertices are not collinear.

Triangles ADB and ADC have equal side halves BD and DC, the common side AD, and equal included right angles. SAS makes them congruent, so corresponding sides give

The two-to-one theorem applies separately to every median in any nondegenerate triangle. The medians can have different whole lengths, but the internal division ratio is fixed.

A specified positive distance from a fixed endpoint along a fixed segment selects one point. The common endpoint and common ray matter; equal distances on different rays would not suffice.

The equal circles touch at the midpoint when each radius is half of BC. A line through two arc intersections requires two distinct points, which this choice does not supply.

The medians depend on the boundary vertices, so the geometric centroid remains fixed. The added weight changes the mass distribution, and the uniform-sheet balance conclusion no longer applies.

The two given side ratios and the vertical angle at G establish SAS similarity of triangles AGB and DGE, with factor 2. Corresponding angles then give DE parallel to AB.

The corresponding side ratio gives

Now triangles CDE and CBA are similar by AA, with factor one half. Therefore CD is half of CB and CE is half of CA. D and E are the two side midpoints.

The vertex portion must be twice the midpoint portion.

The last equality is false, so no real x satisfies the ratio. In particular, no positive lengths satisfy it.

Two correctly constructed medians have exactly one intersection, the centroid. Both passing through P identifies P with that point. Being inside or lying on arbitrary vertex-to-side segments is insufficient.