calculus - Proving and deriving a Gamma function

I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula:

a.) Prove that

$$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = 2\int_0^{1/2}x^{p-1}(1-x)^{p-1}\mathrm dx$$

b.) Make a suitable change of variable (a) and derive the duplication formula for the Gamma function:

$$\Gamma (2p)\Gamma\left(\frac12\right) = 2^{2p-1}\Gamma (p)\Gamma\left(p-\frac12\right)$$