Instructions

PQR is a triangle
[display]O\in PQR[/display]
[display]d_1//PQ \wedge O\in d_1[/display]
[display]d_2//QR \wedge O\in d_2[/display]
[display]d_3//QR \wedge O\in d_3[/display]
WHat's the ratio between the area of the 3 small triangles and PQR ? (Greyed in figure)

Dumy solution

Really, I don't know a dumy solution ! Please, give me one ;-)

Smart solution (click to open)

Look at diagrams bellow :
We appended the image of himself then skewed it a bit.

The side of each triangle is the square root of the area of each surface !

We get as answer: [display]\sqrt{S_{yellow}}+\sqrt{S_{blue}}+\sqrt{S_{green}}=\sqrt{S_{total}}[/display] Yep, this is a rectangle and I use square root to find the side !

But's because rectangle and square differ only by length.

The side of each triangle is the square root of the area of each surface !

We get as answer: [display]\sqrt{S_{yellow}}+\sqrt{S_{blue}}+\sqrt{S_{green}}=\sqrt{S_{total}}[/display] Yep, this is a rectangle and I use square root to find the side !

But's because rectangle and square differ only by length.