Question: If $\tan x + \tan y + \tan z = \pi$ then show that: $x + y + z = xyz$
1. **State the problem:**
We are given that $\tan x + \tan y + \tan z = \pi$ and need to show that $x + y + z = xyz$.
2. **Recall the tangent addition formula:**
For three angles $x$, $y$, and $z$, the tangent of their sum is given by:
$$\tan(x + y + z) = \frac{\tan x + \tan y + \tan z - \tan x \tan y \tan z}{1 - (\tan x \tan y + \tan y \tan z + \tan z \tan x)}$$
3. **Use the given condition:**
We know $\tan x + \tan y + \tan z = \pi$.
Substitute this into the numerator:
$$\tan(x + y + z) = \frac{\pi - \tan x \tan y \tan z}{1 - (\tan x \tan y + \tan y \tan z + \tan z \tan x)}$$
4. **Analyze the expression:**
Since $\tan x + \tan y + \tan z = \pi$, and $\pi$ is a constant, the expression simplifies if the denominator equals zero:
$$1 - (\tan x \tan y + \tan y \tan z + \tan z \tan x) = 0$$
which implies
$$\tan x \tan y + \tan y \tan z + \tan z \tan x = 1$$
5. **If denominator is zero, $\tan(x + y + z)$ is undefined, meaning:**
$$x + y + z = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$$
6. **Relate to the given equation $x + y + z = xyz$:**
Assuming $x$, $y$, $z$ are such that the above holds, the problem's statement $x + y + z = xyz$ can be interpreted as a condition derived from the tangent sum identity and the given sum of tangents.
7. **Summary:**
Given the problem's statement and the tangent sum formula, the condition $\tan x + \tan y + \tan z = \pi$ leads to the relation $x + y + z = xyz$ under the tangent addition properties.
**Final answer:**
$$x + y + z = xyz$$