Wikipedia shows how to generate Triangular-distributed random variates using a variate $U$ drawn from the uniform distribution.

A "Double Triangular" distribution is a special case of a mixture of two triangular distributions. Specifically, it is determined by three numbers $a \lt c \lt b$ and a proportion $p$ with $0 \lt p \lt 1$. It is supported on the interval $[a, b]$.

On the interval $[a,c]$ its density function is given by

$$f(x) = \lambda(x-a)$$

where $\lambda(c-a)^2 = 2p$.

On the interval $[c, b]$ its density function is given by

$$f(x) = \mu(b-x)$$

where $\mu(b-c)^2 = 2(1-p)$.

What would be a good approach to generating Double-Triangular-distributed random variates?