When proving that triangles ABC are ADB are similar it is clear since both contain a right angle and share the angle BAD. We have known shown that these are similar since they are equiangular and it follows of course that the final angle is also equal due to the interior angle sum of a triangle.
When proving that triangles ABC and BDC are similar it is again very clear since we they both contain a right angle and share the angle BCD implying once again that they are equiangular.
It follows from this that triangles ADB and BDC are also similar since they both share similarity with triangle ABC