Statistical Inference — estimating the Cauchy location parameter

Let $x_1,x_2,…,x_n$ be a random sample from a Cauchy distribution having pdf $f(x,\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x,\theta<\infty$. Consider the following statements: 1. $\bar X$ is unbiased for $\theta$ but not consistent estimator for $\theta$. 2. Sample median is consistent estimator for $\theta$. Which of the statements given above is/are correct?