How is the solution for \$\alpha\$ and \$\beta\$ current gain of a BJT derived

bjtcurrentgainmath

I'm just an enthusiast, and not too good at maths; I'm exploring constant current source circuits, in doing so I came across two equations for BJT current gain:

Could someone help me understand how \$\alpha\$ and \$\beta\$ are derived to the equations below:

\$\alpha=\frac{\beta}{\beta+1}\$ and \$\beta=\frac{\alpha}{1-\alpha}\$

I know that:

\$I_c=\alpha I_e\$ and \$I_c=\beta I_b\$

Any help would be much appreciated.

Thanks
Alex