Definition: In geometry, the median of a triangle is a line segment joining a vertex and the midpoint of the opposite side.

Regulations:

1. Every triangle has exactly three medians.

Interactivity

- Click and drag any vertex of the triangle.
- Click the checkboxes to show/hide the different medians.

2. Every median divides the triangle into two smaller triangles of equal area.

Interactivity - A_{\triangle \text{ABAm}} = A_{\triangle \text{ACAm}}%%

- Click and drag any vertex of the triangle.

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Proof

Open the interactivity above. We want to show: A_{\triangle \text{ABAm}} = A_{\triangle \text{ACAm}}%% .

- \triangle \text{ABAm} и \triangle \text{ACAm} have the same height h .
- The area of the triangle \triangle \text{ABAm} is: A_{\triangle \text{ABAm}} = \frac{1}{2} h \cdot \overline{BAm} .
- The area of the triangle \triangle \text{ACAm} is: A_{\triangle \text{ACAm}} = \frac{1}{2}\, h \cdot \overline{CAm} .
- Because \overline{АAm} is a median, the bases are the same, i.e. \overline{BAm}=\overline{BAm} .
- It follows: \frac{1}{2} h \cdot \overline{BAm} = \frac{1}{2}\, h \cdot \overline{CAm} , that is
- A_{\triangle \text{ABAm}} = A_{\triangle \text{ACAm}}%% , as we wanted.

3. The three medians intersect in a common point T called the centroid of the triangle.

Interactivity

- Click and drag any vertex of the triangle.

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Discussion about proof

- The proof is a direct application of Ceva's theorem and the definition of median.

- From this it follows that the centroid is always an internal point of the triangle.

4. The three medians divide the triangle into 6 smaller triangles of equal area.

Interactivity

- Click and drag any vertex of the triangle. Click the checkboxes to show/hide the various triangles.

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Proof

1. Using the same proof as in regulation 2 (above), the areas of T1 and T2 are equal, the areas of T3 and T4 are equal and the areas of T5 and T6 are equal.

2. It remains to show that А_{T1} = А_{T3} = А_{T5}

3. According to regulation 2, the areas of the triangles ABAm and ACAm are equal.

4. Substitution from (2) into (1), we have А_{T1}+А_{T1}+А_{T3} = А_{T3}+А_{T5}+А_{T5} so that

Conclusion All six little triangles have the same area.

Interactivity

- Click and drag any vertex of the triangle. Notice how the part of the median towards the vertex is always 2 times longer than the part towards the midpoint. (If it is not "exactly" twice, increase the number of decimals through the command Options -> Decimal places.)

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