Exercise $1$: Straight beam: Statically Indeterminate

A beam defined in figure below, is submitted to a single load $P$ at the middle $($L$/2).$ The characteristics of the problem are:

Simple support at point A $($Vertical and horizontal displacements null$)$

Support at points B and C $($only vertical displacement null$)$

Lengths between points defined into the figure: $($AD$),($DC$),($CB$)$

Young Modulus E and area moment of inertia I

Define:

Demonstrate that the system is statically indeterminate

Reactions at points A$,$ B and C $($use method of superposition$)$

By applying unit load method $($principle virtual of work$)$:

Vertical deflection at point D $($location of applied force P$)$