Why is the mean of a Chi Square distribution equal to the degree of freedom? Is this part of the definition of a Chi Square distribution?
Or is it deduced from another property? In which case, what is the proof?
 A: You don't define the mean to be the degrees of freedom (d.f.) -- it follows from the definition of the pdf and the definition of expectation of a random variable. 
The pdf of a chi-squared random variable with $k$ d.f. is:
${\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\; x>0, k>0$  (and $0$ elsewhere)
The expectation of a continuous random variable is:
$\operatorname {E} [X]=\int _{-\infty }^{\infty }xf(x)\,\mathrm {d} x.$
So the mean of a chi-square random variable is:
$\operatorname {E} [X]=\int _{0}^{\infty }x{\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,\mathrm {d} x $
Pulling the constants out and combining the $x$ powers
${\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\; \int _{0}^{\infty }x^{{\frac {k+2}{2}}-1}e^{-{\frac {x}{2}}}\,\mathrm {d} x $
the term in the integral can be recognized as another chi-square (missing the normalizing constant). If you multiply and divide by the relevant normalizing constant so that the integral is 1, you're left with a ratio of normalizing constants out the front (for different d.f.) ... which you should be able to simplify.

However, if your question is really "why choose that pdf to be called a chi-square?", whuber's comment is relevant -- the sum of squares of independent standard normals is a random variable that fairly naturally arises in a number of contexts*, and that is something we would therefore like to have a name for. The degrees of freedom relates to the number of independent normals involved and each of those squared components has mean 1. 
* So for example, Helmert identified it as related to the distribution of sample variance for iid samples from a normal distribution (though the use of the symbol $\chi^2$ and hence the name "chi-squared" don't come until Pearson's work about a generation later). 
A: Chi squared distribution is Gamma distribution with parameters $(\lambda, \alpha) = (\frac12, \frac{n}2)$.
And we know that expected value of Gamma is $\frac{\alpha}{\lambda}$ .
Hence the expected value of a chi squared r.v. is $ \frac{n/2}{1/2} = n$.
