Connecting the points $A$ to $C_1$, $B$ to $D_1$, $C$ to $A_1$ and $D$ to $B_1$, we have
$$\begin{align*} P_{\Delta ABC} &= P_I = P_{II} \\ P_{\Delta BCD} &= P_{III} = P_{IV} \\ P_{\Delta CDA} &= P_V = P_{VI} \\ P_{\Delta DAB} &= P_{VII} = P_{VIII} \end{align*}$$
where

$$\begin{align*} P_I &= P_{\Delta BA_1C}, \quad P_{II} = P_{\Delta CA_1B_1}, \quad P_{III} = P_{\Delta DCB_1}, \quad P_{IV} = P_{\Delta C_1DB_1}, \\ P_V &= P_{\Delta C_1DA}, \quad P_{VI} = P_{\Delta C_1D_1A}, \quad P_{VII} = P_{\Delta AD_1B}, \quad P_{VIII} = P_{\Delta BD_1A_1} \end{align*}$$
Now
$$$$
$$\begin{gather*} P_{\Delta ABC} + P_{\Delta BCD} + P_{\Delta CDA} + P_{\Delta DAB} = P_I + P_{III} + P_V + P_{VII} = P_{II} + P_{IV} + P_{VI} + P_{VIII} = 2P \\ P_I = P + (P_I + P_{II} + P_{III} + P_{IV} + P_V + P_{VI} + P_{VII} + P_{VIII}) = P + (2P + 2P) = 5P \end{gather*}$$