I'm looking for a help for drawing this diagrams

Question

I'm looking for a help for drawing this diagrams:

Answer 1

For the second figure, hope the following code helps. I cannot clearly see whether one of the curves is dashed. If that's not the case, remove [dashed] in the code.

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\coordinate (o) at (0,2);
\coordinate (a) at (-4,0);
\coordinate (b) at (4,0);
\draw (o) -- (a) -- (b) -- cycle;
 \draw (0,0) arc (-53:53:1.25);
\draw[dashed] (0,2) arc (-53:53:-1.25);
\end{tikzpicture}

 \end{document}

As for the first figure, you can easily achive with tikz-cd. This is one possibility.

\documentclass{standalone}
\usepackage{tikz}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=small]
 \mathcal{X}^*\arrow{rr}{\rho} &  & \mathcal{M}^*\\
 & \frac{\mathcal{X}^*}{\mathcal{M}^\perp} \arrow[swap]{ur}{\tilde{\rho}}\arrow{ul}{Q}&
 \end{tikzcd}

\end{document}

with the following result:

If you want to keep the fonts (almost) in your image,

\documentclass{standalone}
\usepackage{tikz}
\usepackage{tikz-cd}
\usepackage{amsmath}
\usepackage[mathscr]{euscript}

\begin{document}
\begin{tikzcd}[column sep=small]
 \mathscr{X}^*\arrow{rr}{\rho} &  & \mathscr{M}^*\\
 & \dfrac{\mathscr{X}^*}{\mathscr{M}^\perp} \arrow[swap]{ur}{\tilde{\rho}}\arrow{ul}{Q}&
 \end{tikzcd}
\end{document}

Answer 2

For the first commutative diagram (still it need some fonts improvements):

\documentclass{standalone}

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}
    \begin{tikzpicture}
        \matrix (m) [matrix of math nodes, row sep=3em,
                column sep=2.5em, text height=1.5ex, text depth=0.25ex] {
            \mathcal{X}^* & & \mathcal{M}^* \\
            & \dfrac{\mathcal{X}^*}{\mathcal{M}^{\perp}} & \\ };
            \path[->] (m-1-1) edge node[above] {$\rho$} (m-1-3);
            \path[->] (m-2-2) edge node[below] {$Q$} (m-1-1);
            \path[->] (m-2-2) edge node[below] {$\tilde\rho$} (m-1-3);
    \end{tikzpicture}
\end{document}

Result: