CTK Insights

01 Feb

Medians in a Triangle Meet at the Center: a Second Look

The medians of a triangle meet at a point known at the center of the triangle. Reversing the problem we may ask a relevant question:

Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the medians. If so, construct the triangle.

Solution

Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the medians. If so, construct the triangle.

Solution

The answer to the question is in positive, and the triangle can be constructed is as follows.

Pick point A on α, find points B on β and C on γ such that AB is bisected by γ and AC is bisected by β. There is a simple way to achieve that goal. (We already used this construction in finding a parallelogram cross-section of a pyramid.)

In ΔABC, β and γ house two of the medians BB' and CC'. The third median AA' meets them at the center of the triangle and lies, therefore, on α, implying that α bisects BC.

Obvioiusly, the construction is not unique, what is unique is the shape of the triangles - they all are similar.

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