각의 이등분선 작도와 합동 추론

문제

다음 그림과 같이 $\triangle ABC$에서 $\angle ABC$의 이등분선을 작도하여 변 $AC$와 만나는 점을 $D$라고 하자. 점 $D$에서 변 $AB$에 내린 수선의 발을 $E$, 변 $BC$에 내린 수선의 발을 $F$라고 할 때, 항상 옳은 것을 보기에서 모두 고른 것은?

\begin{center}
\begin{tikzpicture}[scale=0.8]
    \coordinate (A) at (-3, 3);
    \coordinate (B) at (0, 0);
    \coordinate (C) at (4, 1);
    \draw (A) node[above left] {$A$} -- (B) node[below] {$B$} -- (C) node[below right] {$C$} -- cycle;

    % Angle bisector of B
    \path (A) -- (B) -- (C);
    \pgfmathanglebetweenpoints{\pgfpointanchor{B}{center}}{\pgfpointanchor{A}{center}}
    \let\angleBA\pgfmathresult
    \pgfmathanglebetweenpoints{\pgfpointanchor{B}{center}}{\pgfpointanchor{C}{center}}
    \let\angleBC\pgfmathresult
    \pgfmathparse{(\angleBA + \angleBC)/2}
    \let\angleBisectorB\pgfmathresult
    \coordinate (D_on_AC) at (intersection of B--($ (B) + (\angleBisectorB:4cm) $) and A--C);
    \node[above] at (D_on_AC) {$D$};
    \draw[dashed] (B) -- (D_on_AC);

    % Perpendicular from D to AB
    \coordinate (E_on_AB) at (projection of D_on_AC on A--B);
    \node[left] at (E_on_AB) {$E$};
    \draw (D_on_AC) -- (E_on_AB);
    \draw pic [draw, angle radius=5mm, angle eccentricity=1.2, angle counterclockwise] {right angle = D_on_AC--E_on_AB--B};

    % Perpendicular from D to BC
    \coordinate (F_on_BC) at (projection of D_on_AC on B--C);
    \node[right] at (F_on_BC) {$F$};
    \draw (D_on_AC) -- (F_on_BC);
    \draw pic [draw, angle radius=5mm, angle eccentricity=1.2, angle counterclockwise] {right angle = D_on_AC--F_on_BC--B};

    % Mark angles
    \draw pic [draw, angle radius=8mm, angle eccentricity=1.2] {angle = A--B--D_on_AC};
    \draw pic [draw, angle radius=8mm, angle eccentricity=1.2] {angle = D_on_AC--B--C};
\end{tikzpicture}
\end{center}

(가) $\triangle DBE \equiv \triangle DBF$
(나) $DE = DF$
(다) $AE = CF$
(라) $AD = CD$