Geometry

[Thibaut25]

Hi guys, I stumbled on this exercice and I could use some help to solve it. It would help me a great deal to pass my exam. I am not sure how to prove it and even how to start. Area or surface, I am not sure wich word to use in the context. I would be really grateful if someone could fine the solution

 

[Dr.Peterson]

Are you saying you are supposed to prove that equality?

Please, as we ask you to, tell us what ideas you have tried, and where you are stuck. Also, what have you learned in the course that this exam is intended to test? We are not here to just give answers, but to help you learn to solve problems yourself.

I suspect it may be useful to consider the fact that, for similar figures, area is proportional to the square of any linear dimension.

 

[Thibaut25]

I am supposed to prove the equation. I was able to prove that all the triangles have the same shape. I then expressed the area in function of two sides and an angle. I used the same angle for the 4 triangles but now I am stuck and cannot go forward.

 

[Dr.Peterson]

Have you thought about my suggestion? Since you know the triangles are similar, their areas are proportional to the squares of their bases (C'P, PB', C'''B''', and CB).

Now, how are those four segments related? (How must they be related if the equation is to be true? Can you then prove that?)

 

[yma16]

If you label the base and the height as a1,a2, a3, h1,h2, and h3, the equality is ((a1+a2+a3)(h1+h2+h3)/2)^.5=... Square both sides and cancel some terms, you will get the result.