To solve the problem with triangle \(ABC\) where \(A\) is the right angle:

1. **Calculate \(AC\)**:
Since \(AB = 3\) cm and \(BC = 5\) cm, we use the Pythagorean theorem:
\[
AC = \sqrt{BC^2 - AB^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ cm}
\]

2. **Show that quadrilateral \(ABMD\) is a cyclic quadrilateral**:
- Since \(M\) is the midpoint of \(BC\), we find its coordinates:
- \(B(0, 3)\), \(C(5, 0)\) \(\Rightarrow M\left(\frac{0+5}{2}, \frac{3+0}{2}\right) = \left(\frac{5}{2}, \frac{3}{2}\right)\)
- Reflect \(M\) across \(AC\) to find \(D\) and verify symmetry properties.

3. **Point \(E\) is the midpoint of \(AB\)**:
- \(E\left(\frac{0+0}{2}, \frac{3+0}{2}\right) = (0, 1.5)\).
- Show that \(F\) (the reflection of \(M\) across \(E\)) maintains symmetry regarding \(E\) and \(M\).

4. **Show that quadrilateral \(AFMC\) is also cyclic**:
- Validate the properties of cyclic quadrilaterals, using angles and lengths to prove \(AFMC\) fits the requirements.

5. **To show \(A; K; F\) are collinear**:
- Establish coordinates and slopes to check the linear alignment of \(A\), \(K\), and \(F\).

Through these steps, you can conclude the congruencies and properties required by the problem.