평행사변형 내 닮은 도형과 선분 비율 추론 문제

문제

그림과 같이 평행사변형 $ABCD$에서 점 $A$를 지나는 직선이 대각선 $BD$와 만나는 점을 $E$, 변 $BC$와 만나는 점을 $F$, 선분 $DC$의 연장선과 만나는 점을 $G$라 하자. $AE:EF = 3:2$ 이고 $BF:FC = 2:1$ 일 때, $AE:EF:FG$의 비율을 구하시오.

$\text{ }$

graph TD
    A[A] --- B[B]
    B --- C[C]
    C --- D[D]
    D --- A

    style A fill:#fff,stroke:#333,stroke-width:2px;
    style B fill:#fff,stroke:#333,stroke-width:2px;
    style C fill:#fff,stroke:#333,stroke-width:2px;
    style D fill:#fff,stroke:#333,stroke-width:2px;

    subgraph Parallelogram ABCD
        A -- diagonal --> C
        B -- diagonal --> D
        A -- line --> E
        E -- line --> F
        F -- line --> G
    end

    E[E];
    F[F];
    G[G];

    A --- D
    D --- C
    C --- B
    B --- A

    A --- E
    E --- F
    F --- G
    B --- D

    linkStyle 0 stroke-width:0px;
    linkStyle 1 stroke-width:0px;
    linkStyle 2 stroke-width:0px;
    linkStyle 3 stroke-width:0px;
    linkStyle 4 stroke-width:0px;
    linkStyle 5 stroke-width:0px;
    linkStyle 6 stroke-width:0px;
    linkStyle 7 stroke-width:0px;

    linkStyle 8 stroke:red,stroke-width:2px; /* Line AFG */
    linkStyle 9 stroke:blue,stroke-width:2px; /* Diagonal BD */

    
    subgraph Hidden 
        X(( ))
    end
    style X fill:#fff,stroke:#fff;
    X --- D
    X --- C
    linkStyle 10 stroke-width:0px;
    linkStyle 11 stroke-width:0px;