Triangle of medians


B.J Kim

Given any triangle, construct a triangle of medians.

In above figure, there is any triangle ABC.

Construct 3 medians.

3 medians of triangle ABC are the segments from the vertex to midpoint of the opposite sides.

Let's construct a new triangle with 3 sides having the lengths of the three medians from the original triangle.

Step 1. Make a parallel line passing through the midpoint F parallel with the median BE.

Step 2. Make a circle center F with radius BE and then label the intersection between the circle and the parallel line H

Step 3. Similarly, make a parallel line passing through c parallel with the AD

Step 4. Construct a circle center c with radius AD, then the point H would be concurrence.

Step 5. We can construct a new triangle CFH with three medians of the original triangle ABC.


Find some relationship between the two triangles.

Are they congruent? No, because their areas are different

The ratio of the area of the original triangle ABC and the triangle of medians is constant as 1.333...or the ratio between constructed triangle and original triangle is constant as 0.75

The parameter of the original triangle is greater than the constructed triangle.That is, the ratio of the parameter of the original triangle to the constructed triangle is greater than 1.

When I compare the ratio of corresponding sides, the ratios are not same. This implies that these two triangles are not similar.

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