Since the diagram is symmetrical about a vertical axis going through C, the diagram can be flipped horizontally, and it should have no physical effect.

=

Since the flipping should have no physical effect, $T_1 = T_2$ since the $T_1$ on the first image becomes $T_2$ on the second, and vice versa

Now, notice the Pythagoras equation holds for the 3 lengths AB, AC, CB, meaning ACB is 90 degrees.

Draw the free body diagram, and solve the equation for static.

\[\frac{1}{\sqrt{2}}(T_1+T_2) = \frac{1}{\sqrt{2}}(2T_1) = \sqrt{2}T_1 = Wg \\ T_1 = T_2 = \frac{1}{\sqrt{2}}Wg\]