That $a=b+c$ is not true in general is well-covered in other answers. There are, however, numerous isosceles trapezoids for which it is true.

In any isosceles trapezoid $TRAP$ as shown with $TP=RA$, $\triangle TPD\cong\triangle RAD$ and $\triangle RAD$ is the reflection image of $\triangle TPD$ over the line through $D$ perpendicular to $\overline{TR}$ and $\overline{AP}$.
Conversely, starting with any $\triangle XYZ$ and reflecting it over the line through $Z$ for which the acute angles formed by this line and $\overline{XZ}$ and $\overline{YZ}$ have equal measure‡ to $\triangle X'Y'Z$ yields an isosceles trapezoid $XX'Y'Y$ with $Z$ at the intersection of the diagonals.
(‡ This restriction on the angles formed by the line and the sides of the triangle ensures that $X$, $Z$, and $Y'$ are collinear and $X'$, $Z$, and $Y$ are collinear, so that $Z$ is the intersection of the diagonals.)
Now, if you start with any $\triangle XYZ$ with a right angle at $Z$, you will get an isosceles trapezoid with perpendicular diagonals.