The medians of a triangle intersect at a point inside it. The total number of…

Concept

Three straight lines enclose a triangle only when they cross one another at three different points: no two of them may be parallel, and all three must not pass through one common point (that is, they must not be concurrent). So in a figure built from n straight lines, the number of triangles is the number of ways of choosing 3 lines, C(n, 3), reduced by every choice that is blocked by parallel lines or by concurrency, and, when the figure is drawn with segments rather than full lines, only those choices count whose three joining segments are actually drawn.

The figure

Let the triangle be ABC, with D, E and F the midpoints of BC, CA and AB. The three medians AD, BE and CF are drawn, and all of them pass through the centroid G.

Application — counting the line triples

1. The completed figure is made of 6 straight lines: the three sides AB, BC, CA and the three medians AD, BE, CF.

2. Number of ways of choosing 3 of these 6 lines: C(6, 3) = 20.

3. Exactly 4 of those 20 choices are concurrent, so they enclose nothing: {AB, CA, AD} meet at A, {AB, BC, BE} meet at B, {BC, CA, CF} meet at C, and the three medians {AD, BE, CF} meet at G.

4. In every remaining choice the three lines cross at three distinct marked points among A, B, C, D, E, F, G, and each joining segment is actually drawn, so that choice encloses one triangle.

5. Triangles = 20 − 4 = 16.

Cross-check — count them by size

6 + 3 + 6 + 1 = 16, exactly matching the line-triple count.

Answer: 16 triangles.