Excircles and areas

[ins-]

Let $I_1, I_2, I_3$ are the tangent points of the incircle of $ABC$ with the sides and $J_1, J_2, J_3$ are its excenters. Prove that ${S_{ABC}}^2=S_{I_1I_2I_3}S_{J_1J_2J_3}$

[Edward_Tur]

$S_{I_1I_2I_3}=\frac {2abc}{(a+b)(b+c)(c+a)}S$, $S_{J_1J_2J_3}=(a+b+c)R$.
Что в этих задачах олимпиадного? И зачем на каждую такую задачу отдельная тема?

[ins-]

Are you sure about the equality for $S_{I_1I_2I_3}$ ? $\frac{S_{I_1I_2I_3}}{S_{ABC}}=\frac{r}{2R}$. About the problems - olympiads have different levels. They may be good for the earlier rounds. I didn't post them in separate topics because they are in some ways different and I don't like posting many problems in a single topic. It is a matter of taste.

[Edward_Tur]

ins- в сообщении #578367 писал(а):

Are you sure about the equality for $S_{I_1I_2I_3}$ ? $\frac{S_{I_1I_2I_3}}{S_{ABC}}=\frac{r}{2R}$

Да, я перепутал - записал формулу для площади треугольника с вершинами в основаниях биссектрис треугольника.

[ins-]

I know that. I posted a problem here to prove an inequality about the area of these two triangles. I have few "open problems" but they seems to be very hard and they are still not solved. If you want you can try them. It is a sample problem topic57024.html. I have no enough experience to solve them. But I like math and when I see something beautiful.

You can see a solution here:

(Оффтоп)

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=48&t=481695

A friend of mine solved the problem. What I like about this problem is it is not very hard but it requires lots of knowledge about the triangle geometry. And the fact is not "super beautiful" but it is "beautiful enough". It is my personal opinion.