In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively.  The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$  The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime.  Find $a+b+c.$
1. **Identify the given angles and side lengths:**
   - In triangle \(ABC\), \(\angle A = 60^\circ\) and \(\angle B = 45^\circ\).
   - The bisector of \(\angle A\) intersects \(\overline{BC}\) at \(T\), and \(AT = 24\).

2. **Determine the remaining angle:**
   - Since the sum of angles in a triangle is \(180^\circ\), we have:
     \[
     \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 60^\circ - 45^\circ = 75^\circ
     \]

3. **Analyze the special right triangles:**
   - Let \(D\) be the foot of the perpendicular from \(T\) to \(AB\). Since \(\angle ATB = 105^\circ\), triangle \(TBD\) is a \(45^\circ-45^\circ-90^\circ\) right triangle, and triangle \(ATD\) is a \(30^\circ-60^\circ-90^\circ\) right triangle.

4. **Calculate the lengths in triangle \(ATD\):**
   - In a \(30^\circ-60^\circ-90^\circ\) triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\).
   - Given \(AT = 24\), the length of \(TD\) (opposite the \(30^\circ\) angle) is:
     \[
     TD = \frac{AT}{2} = \frac{24}{2} = 12
     \]
   - The length of \(AD\) (opposite the \(60^\circ\) angle) is:
     \[
     AD = TD \cdot \sqrt{3} = 12\sqrt{3}
     \]

5. **Calculate the area of triangle \(ATD\):**
   - The area of triangle \(ATD\) is:
     \[
     \text{Area}_{ATD} = \frac{1}{2} \times AD \times TD = \frac{1}{2} \times 12\sqrt{3} \times 12 = 72\sqrt{3}
     \]

6. **Calculate the lengths in triangle \(TBD\):**
   - Since \(TBD\) is a \(45^\circ-45^\circ-90^\circ\) triangle, \(BD = TD = 12\).

7. **Calculate the area of triangle \(TBD\):**
   - The area of triangle \(TBD\) is:
     \[
     \text{Area}_{TBD} = \frac{1}{2} \times BD \times TD = \frac{1}{2} \times 12 \times 12 = 72
     \]

8. **Analyze triangle \(ATC\):**
   - Let \(E\) be the foot of the perpendicular from \(T\) to \(AC\). Since \(\angle ATE = 60^\circ\), triangle \(ATE\) is a \(30^\circ-60^\circ-90^\circ\) right triangle.
   - Given \(AT = 24\), the length of \(TE\) (opposite the \(30^\circ\) angle) is:
     \[
     TE = \frac{AT}{2} = \frac{24}{2} = 12
     \]

9. **Determine the isosceles property of triangle \(ATC\):**
   - Since \(\angle ATC = \angle ACT = 75^\circ\), triangle \(ATC\) is isosceles with \(AT = AC = 24\).

10. **Calculate the area of triangle \(ATC\):**
    - The area of triangle \(ATC\) is:
      \[
      \text{Area}_{ATC} = \frac{1}{2} \times AC \times TE = \frac{1}{2} \times 24 \times 12 = 144
      \]

11. **Sum the areas to find the total area of triangle \(ABC\):**
    - The total area of triangle \(ABC\) is:
      \[
      \text{Area}_{ABC} = \text{Area}_{TBD} + \text{Area}_{ATD} + \text{Area}_{ATC} = 72 + 72\sqrt{3} + 144 = 216 + 72\sqrt{3}
      \]

12. **Identify \(a\), \(b\), and \(c\):**
    - The area of triangle \(ABC\) is in the form \(a + b\sqrt{c}\), where \(a = 216\), \(b = 72\), and \(c = 3\).

13. **Calculate \(a + b + c\):**
    - Therefore, \(a + b + c = 216 + 72 + 3 = 291\).

The final answer is \(\boxed{291}\)