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\begin{document}

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\begin{center}
{\Large NMU Math \& CS Department} \\
\vspace{0.5cm}
{\Large Problem of the Month, October 2022}
\end{center}

\vspace{1cm}

\paragraph{} Suppose that $\alpha,\beta,\gamma$ are positive angles with $\alpha+\beta+\gamma=\displaystyle \frac{\pi}{2}$.

\paragraph{} Express the unknown lengths in the picture below in such a manner that proves:

\begin{equation*}
\text{tan}(\alpha)\text{tan}(\beta)+\text{tan}(\beta)\text{tan}(\gamma)+\text{tan}(\alpha)\text{tan}(\gamma)=1.	
\end{equation*}

\vspace{0.5cm}

\begin{center}
\begin{tikzpicture}[scale=1.5] 
\coordinate (A) at (0,7) {};
\coordinate (B) at (5,7) {};
\coordinate (C) at (5,0) {};
\coordinate (O) at (0,0) {};
\coordinate (D) at (2.6,7) {};
\coordinate (E) at (5,4) {};
\draw (O) -- (A) -- (B) -- (C) --node[below,yshift=-3mm]{$1$} (O);
\draw (O) -- (D) -- (E) -- (O);
\tkzMarkRightAngle[size=.25](O,A,B);
\tkzMarkRightAngle[size=.25](O,E,D);
\tkzMarkRightAngle[size=.25](A,B,C);
\tkzMarkRightAngle[size=.25](O,C,E);
\tkzMarkAngle[mark=none,arc=l,size=1.5cm,line width=0.75pt,arrows=->](C,O,E);
\tkzMarkAngle[mark=none,arc=l,size=1.5cm,line width=0.75pt,arrows=->](E,O,D);
\tkzMarkAngle[mark=none,arc=l,size=1.5cm,line width=0.75pt,arrows=->](D,O,A);
\tkzLabelAngle[dist=1.2](C,O,E){$\alpha$};
\tkzLabelAngle[dist=1.2](E,O,D){$\beta$};
\tkzLabelAngle[dist=1.2](D,O,A){$\gamma$};
\end{tikzpicture}
\end{center}

\end{document}