Lines have been drawn such that the 1/3 points of each side meet their opposite vertex.
If the area shaded green is 63cm^{2}, what is the total area of the triangle ABC?

Thanks for your reply. The GeoGebra demo is very neat, but I don’t think it is near to a proof, is it? I think it is just a demonstration of something we know to be true. For example, how do we know that when we join a vertex A of the inner triangle ABC to a midpoint of a ‘nearby’ side of the outer triangle, that the resulting line segment is parallel to the side BC ?

Area of triangle formula as in 1/2 x b x h or 1/2 ab sin C?

it’s a right angled triangle so they’re one in the same.

I’ve played around in Geogebra and got the relationship between the area of the white part and the whole triangle, and then got 73.5 as an answer… cheating, no?

Thanks for your reply. The GeoGebra demo is very neat, but I don’t think it is near to a proof, is it? I think it is just a demonstration of something we know to be true. For example, how do we know that when we join a vertex A of the inner triangle ABC to a midpoint of a ‘nearby’ side of the outer triangle, that the resulting line segment is parallel to the side BC ?

it is in fact a proof. You can find it in:

W. Johnston and J. Kennedy, Heptasection of a triangle, Mathematics Teacher, 8b (1993), p.192

Is it 73.5? (I actually make it 73.49999, but that’s floating point for you)

it is 🙂 Well done.

It might seem really obvious, but where do I begin with this?

Start messing with pythagoras formula, area of triangle formula and perimeter formula

Area of triangle formula as in 1/2 x b x h or 1/2 ab sin C?

it’s a right angled triangle so they’re one in the same.

I’ve played around in Geogebra and got the relationship between the area of the white part and the whole triangle, and then got 73.5 as an answer… cheating, no?

ICYMI, our non-Geogebra, non-calculator proof:

That’s rxcell

In case of overlap triangles in a triangle,

Area of triangle ABC =63 sq cm,

*excellent