This problem appeared:
If the points (a,2a) (2a,a), (a,a)and enclose a triangle of area 18 sq units, the centroid of the
triangle is ?
Let's solve it with geogebra!
1)Create a slider called a and range it from 0 to 10.
2) Enter points (a,a),(a,2a)(2a,a).
3) Use the polygon tool to create a triangle with those 3 points as the vertices.
4) Get the midpoints of the 3 sides of the triangle using the midpoint tool.
5) Draw line segments from each vertices to the opposite midpoint.
6) Get the intersection of those 3 line segments and call it point G. That is the centroid.
7) Move the slider until the triangle has an area of 18. You should be able to get it exactly.
8) Read off the coordinates of the centroid.
What did you get? If we assume a is positive is that the only answer?
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.