Because what you propose is impossible. You are essentially trying to make a cycle out of only these three steps:

1. Isothermal expansion (A to B)

2. Adiabatic expansion (B to C)

3. Adiabatic compression back to original state (C to A)

The curve going from C to A cannot be an adiabatic process. Adiabatic processes are characterized by $$PV^n=\text{const}$$ where $n$ is a property of the gas being used.

Therefore, if you want to follow an adiabatic curve during compression, you will just end up going back to state B. You can't go to state A from C using an adiabatic compression.

To be a little more specific, let's say the pressure and volume at states $B$ and $C$ are $(P_B,V_B)$ and $(P_C,V_C)$ respectively. Then we know in process 2 $$P_BV_B^n=P_CV_C^n=\alpha$$ Or, in other words, the entire curve is described by $$P=\frac{\alpha}{V}=\frac{P_BV_B^n}{V}=\frac{P_CV_C^n}{V}$$ Now we want to do adiabatic compression from state C. Well we have to follow the curve defined by $PV^n=\beta$, but since we know we start in state $C$ it must be that the constant is the same one as before: $\beta=\alpha=P_CV_C^n$. Therefore, the curve is given by $$P=\frac{P_CV_C^n}{V}$$ which is the same curve we followed going from B to C.

Because what you propose is impossible. You are essentially trying to make a cycle out of only these three steps:

1. Isothermal expansion (A to B)

2. Adiabatic expansion (B to C)

3. Adiabatic compression back to original state (C to A)

The curve going from C to A cannot be an adiabatic process. Adiabatic processes are characterized by $$PV^n=\text{const}$$ where $n$ is a property of the gas being used.

Therefore, if you want to follow an adiabatic curve during compression, you will just end up going back to state B. You can't go to state A from C using an adiabatic compression.

This is why we need the isothermal compression step after the adiabatic expansion step. This step is needed so that we can get on the correct adiabatic curve back to state A

To be a little more specific, let's say the pressure and volume at states $B$ and $C$ are $(P_B,V_B)$ and $(P_C,V_C)$ respectively. Then we know in process 2 $$P_BV_B^n=P_CV_C^n=\alpha$$ Or, in other words, the entire curve is described by $$P=\frac{\alpha}{V^n}=\frac{P_BV_B^n}{V^n}=\frac{P_CV_C^n}{V^n}$$ Now we want to do adiabatic compression from state C. Well we have to follow the curve defined by $PV^n=\beta$, but since we know we start in state $C$ it must be that the constant is the same one as before: $\beta=\alpha=P_CV_C^n$. Therefore, the curve is given by $$P=\frac{\beta}{V^n}=\frac{P_CV_C^n}{V^n}$$ which is the same curve we followed going from B to C.

We need the isothermal compression step in order to get to the appropriate state D such that $P_DV_D^n=P_AV_A^n$

Because what you propose is impossible. You are essentially trying to make a cycle out of only these three steps:

1. Isothermal expansion (A to B)

2. Adiabatic expansion (B to C)

3. Adiabatic expansion back to original state (C to A)

The curve going from C to A cannot be an adiabatic process. Adiabatic processes are characterized by $$PV^n=\text{const}$$ where $n$ is a property of the gas being used.

Therefore, if you want to follow an adiabatic curve during compression, you will just end up going back to state B. You can't go to state A from C using an adiabatic compression.

Because what you propose is impossible. You are essentially trying to make a cycle out of only these three steps:

1. Isothermal expansion (A to B)

2. Adiabatic expansion (B to C)

3. Adiabatic compression back to original state (C to A)

The curve going from C to A cannot be an adiabatic process. Adiabatic processes are characterized by $$PV^n=\text{const}$$ where $n$ is a property of the gas being used.

Therefore, if you want to follow an adiabatic curve during compression, you will just end up going back to state B. You can't go to state A from C using an adiabatic compression.

To be a little more specific, let's say the pressure and volume at states $B$ and $C$ are $(P_B,V_B)$ and $(P_C,V_C)$ respectively. Then we know in process 2 $$P_BV_B^n=P_CV_C^n=\alpha$$ Or, in other words, the entire curve is described by $$P=\frac{\alpha}{V}=\frac{P_BV_B^n}{V}=\frac{P_CV_C^n}{V}$$ Now we want to do adiabatic compression from state C. Well we have to follow the curve defined by $PV^n=\beta$, but since we know we start in state $C$ it must be that the constant is the same one as before: $\beta=\alpha=P_CV_C^n$. Therefore, the curve is given by $$P=\frac{P_CV_C^n}{V}$$ which is the same curve we followed going from B to C.