Triangle `ABC` has `AC=13, AB = 15 and BC = 14.` Let `'O'` be the circumcentre of the `DeltaABC.` If the length of perpendicular from the point `'O'` on `BC` can be expressed as a rational `m/n` in the lowest form then find `(m +n).`

Answer

To solve the problem step-by-step, we will follow the outlined approach to find the length of the perpendicular from the circumcenter \( O \) to the side \( BC \) in triangle \( ABC \).

### Step 1: Identify the sides of the triangle
Given:
- \( AC = 13 \)
- \( AB = 15 \)
- \( BC = 14 \)

Let:
- \( a = BC = 14 \)
- \( b = AC = 13 \)
- \( c = AB = 15 \)

### Step 2: Calculate the semi-perimeter \( S \)
The semi-perimeter \( S \) is calculated as:
\[ S = \frac{a + b + c}{2} = \frac{14 + 13 + 15}{2} = \frac{42}{2} = 21 \]

### Step 3: Calculate the area \( \Delta \) using Heron's formula
Heron's formula states that the area \( \Delta \) is given by:
\[ \Delta = \sqrt{S(S-a)(S-b)(S-c)} \]
Substituting the values:
\[ \Delta = \sqrt{21 \times (21 - 14) \times (21 - 13) \times (21 - 15)} = \sqrt{21 \times 7 \times 8 \times 6} \]

### Step 4: Simplify the area calculation
Calculating the product:
\[ \Delta = \sqrt{21 \times 7 \times 8 \times 6} = \sqrt{21 \times 7 \times 48} \]
Breaking it down:
\[ = \sqrt{147 \times 48} \]
Calculating \( 147 \) and \( 48 \):
\[ 147 = 7 \times 21, \quad 48 = 16 \times 3 \]
Thus,
\[ \Delta = \sqrt{7 \times 21 \times 16 \times 3} = \sqrt{7 \times 3 \times 336} = \sqrt{7056} = 84 \text{ square units} \]

### Step 5: Calculate the circumradius \( R \)
The circumradius \( R \) is given by:
\[ R = \frac{abc}{4\Delta} \]
Substituting the values:
\[ R = \frac{14 \times 13 \times 15}{4 \times 84} \]

### Step 6: Calculate \( R \)
Calculating \( abc \):
\[ abc = 14 \times 13 \times 15 = 2730 \]
Now substituting into \( R \):
\[ R = \frac{2730}{336} = \frac{65}{8} \]

### Step 7: Calculate \( \cos A \)
Using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values:
\[ \cos A = \frac{13^2 + 15^2 - 14^2}{2 \times 13 \times 15} = \frac{169 + 225 - 196}{390} = \frac{198}{390} = \frac{99}{195} = \frac{33}{65} \]

### Step 8: Calculate the length of the perpendicular from \( O \) to \( BC \)
The length of the perpendicular from \( O \) to \( BC \) is given by:
\[ OM = R \cos A = \frac{65}{8} \times \frac{33}{65} = \frac{33}{8} \]

### Step 9: Express \( OM \) in the form \( \frac{m}{n} \)
Here, \( m = 33 \) and \( n = 8 \).

### Step 10: Find \( m + n \)
Thus,
\[ m + n = 33 + 8 = 41 \]

### Final Answer
The final answer is:
\[ \boxed{41} \]