Help geo mains or if u're good at geo

[frosty flax]

do you have the original problem, by any chance?

[final vessel]

No srry

Is there smth wrong with the illustration?

[wary prism]

Hint:|| just rotate each smaller triangle by 60 degrees around its corresponding vertex A,B,C.||

[frosty flax]

i don't think so, as someone came up with a hint

[wary prism]

||S is an area of ABC. After the rotation the area of the hexagon equals two areas of ABC. On the other hand the hexagon consists of 3 equilateral triangles with sides 5,12,13 plus three equal right triangles with sides 5,12,13.||

[final vessel]

Ohh that's genius

Is there a theorem that suggest that the hexagon is equals to 2•triangle?

[wary prism]

you don't need a theorem. Just by construction.

The hexagon consists of the triangle ABC plus copies of its parts (rotated). So, they make 2S.

[boreal bay]

Is the answer ||(180+169√3)/4||?

[foggy nexus]

That’s ugly…..but might be the correct answer

[wary prism]

yes

[boreal bay]

How can I make it prettier?

[foggy nexus]

它已經是答案了😂😂

I believe it’s the simplest form

[wary prism]

you can divide 180/4=45. One digit less 🙂

[boreal bay]

ok

How to solve if the number isn't 5, 12, 13?

[wary prism]

same as I show in the picture

just use Heron's for the area of the triangle

[boreal bay]

thanks