theorems - francois-rozet/sleek-template GitHub Wiki

Theorems

Theorems features of the sleek-theorems package.

Settings

  • Renewing of \qedsymbol to a black square.

    \renewcommand{\qedsymbol}{$\blacksquare$}

Environments

  • thm, lem, prop, proof, defn, hyp, meth, quest, answ, expl, rmk, note and tip are theorem-like environments with slightly different display styles.

    \begin{env}[arg*]
    ...
\end{env}

  • framedthm, framedlem, framedprop, framedprf, frameddefn, framedhyp, framedmeth, framedquest, framedansw and framedexpl are framed environments with theorem titles.

    \begin{framedenv}[arg*]
    ...
\end{framedenv}

    They all have an unnumbered version framedenv*.

    \begin{framedenv*}
    ...
\end{framedenv*}

arg1* is the optional given name.

Examples

\begin{thm}[Triangle inequality]
    Let be a triangle in Euclidean space. Then the sum of the lengths of two of its sides always surpass or equals the length of the third.
\end{thm}

\begin{proof}
    Let $a$, $b$ and $c$ be the lengths of the sides of a triangle in Euclidean space and $\alpha$, $\beta$, $\gamma$ their respective opposite angle. By the generalized Pythagoras' theorem, we have
    \begin{alignat*}{2}
                              &  & c^2 & = a^2 + b^2 - 2ab \cos\gamma \\
                              &  &     & \leq a^2 + b^2 + 2ab         \\
                              &  &     & \leq (a + b)^2               \\
        \Leftrightarrow \quad &  & c   & \leq a + b
    \end{alignat*}
    Therefore in any triangle, the sum of the lengths of two sides always surpass or equals the length of the third.
\end{proof}

\begin{framedthm}[Triangle inequality]{\label{thm:Triangle inequality}}
    Let be a triangle in Euclidean space. Then the sum of the lengths of two of its sides always surpass or equals the length of the third.
\end{framedthm}

\begin{framedprf}
    Let $a$, $b$ and $c$ be the lengths of the sides of a triangle in Euclidean space and $\alpha$, $\beta$, $\gamma$ their respective opposite angle. By the generalized Pythagoras' theorem, we have
    \begin{alignat*}{2}
                              &  & c^2 & = a^2 + b^2 - 2ab \cos\gamma \\
                              &  &     & \leq a^2 + b^2 + 2ab         \\
                              &  &     & \leq (a + b)^2               \\
        \Leftrightarrow \quad &  & c   & \leq a + b
    \end{alignat*}
    Therefore in any triangle, the sum of the lengths of two sides always surpass or equals the length of the third. \qedadd
\end{framedprf}

\begin{framedquest*}
    Based on the theorem \ref{thm:Triangle inequality}, what is the shortest path from a point $A$ to a point $B$ in Euclidean geometry ?
\end{framedquest*}
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