Excircles

[ins-]

Let $k_a$, $k_b$, $k_c$ are the excircles of the triangle $ABC$ (tangent to BC, CA, AB, respectively). $k_1(r_1)$ is a circle, tangent to $k_a$ and the lines $AB$ and $BC$. $k_2(r_2)$ is a circle, tangent to $k_a$ and the lines $BC$ and $AC$. $k_3(r_3)$ is a circle, tangent to $k_b$ and the lines $AC$ and $BC$. $k_4(r_4)$ is a circle, tangent to $k_b$ and the lines $BC$ and $AB$. $k_5(r_5)$ is a circle, tangent to $k_c$ and the lines $AB$ and $AC$. $k_6(r_6)$ is a circle, tangent to $k_c$ and the lines $AB$ and $BC$. Prove that: $r_1r_3r_5=r_2r_4r_6$.

[Slip]

There are 2 circles tangent to, for example, $AB$, $BC$ and $k_a$, which of them is $k_1$? If all circles are "nearest to the vertex", then it's enough to compute $r_i$ using $r_a$, $r_b$, $r_c$ and the angles of the triangle, which is quite simple

[ins-]

Smaller circles I meant, excuse me.

[ins-]

This problem is not easy and not hard. It can be solved with or without trigonometry.

[nnosipov]

Mechanical proving with complex numbers works too.

[ins-]

I think it can be done easier - my idea was to create a Sangaku-like problem with level of difficulty - regional round of bulgarian olympiad.

[Sergic Primazon]

ins- в сообщении #812863 писал(а):

Let $k_a$, $k_b$, $k_c$ are the excircles of the triangle $ABC$ (tangent to BC, CA, AB, respectively). $k_1(r_1)$ is a circle, tangent to $k_a$ and the lines $AB$ and $BC$. $k_2(r_2)$ is a circle, tangent to $k_a$ and the lines $BC$ and $AC$. $k_3(r_3)$ is a circle, tangent to $k_b$ and the lines $AC$ and $BC$. $k_4(r_4)$ is a circle, tangent to $k_b$ and the lines $BC$ and $AB$. $k_5(r_5)$ is a circle, tangent to $k_c$ and the lines $AB$ and $AC$. $k_6(r_6)$ is a circle, tangent to $k_c$ and the lines $AB$ and $BC$. Prove that: $r_1r_3r_5=r_2r_4r_6$.

$$\frac{r_1}{r_6}=\frac{r_a}{r_c} , \frac{r_3}{r_2}=\frac{r_b}{r_a} , \frac{r_5}{r_4}=\frac{r_c}{r_b}$$

[ins-]

I have one more question. How can we express the radii of a circle tangent to exactly two excircles of a triangle and the incircle by triangle any triangle elements? It is also interesting how can we construct such a circle.

The circles I mentioned are in some way similar to the Feuerbach circle.

[ins-]

http://en.wikipedia.org/wiki/Problem_of_Apollonius I suppose there exist many circles tangent to two excircles and the incircle and it makes my question not defined as it must be.