Suppose a, b, c are three non-coplanar vectors, then
`((a+b+c)*((a+c) times(a+b)))/([a b c])="______"`

Answer

To solve the given problem, we need to evaluate the expression:

\[ \frac{(a+b+c) \cdot ((a+c) \times (a+b))}{[a \, b \, c]} \]

where \( [a \, b \, c] \) represents the scalar triple product of the vectors \( a, b, c \).

### Step 1: Expand the Cross Product
First, we need to compute the cross product \( (a+c) \times (a+b) \).

Using the distributive property of the cross product, we have:

\[ (a+c) \times (a+b) = a \times a + a \times b + c \times a + c \times b \]

Since \( a \times a = 0 \) (the cross product of any vector with itself is zero), we can simplify this to:

\[ (a+c) \times (a+b) = 0 + a \times b + c \times a + c \times b = a \times b + c \times a + c \times b \]

### Step 2: Substitute Back into the Expression
Next, we substitute this result back into our original expression:

\[ \frac{(a+b+c) \cdot (a \times b + c \times a + c \times b)}{[a \, b \, c]} \]

### Step 3: Distribute the Dot Product
Now we will distribute the dot product:

\[ (a+b+c) \cdot (a \times b + c \times a + c \times b) = (a+b+c) \cdot (a \times b) + (a+b+c) \cdot (c \times a) + (a+b+c) \cdot (c \times b) \]

### Step 4: Evaluate Each Dot Product
1. **First Term**: \( (a+b+c) \cdot (a \times b) \)

The dot product of a vector with a cross product of two other vectors is zero if the vector is one of the two vectors being crossed. Thus, \( a \cdot (a \times b) = 0 \) and \( b \cdot (a \times b) = 0 \). Therefore:

\[ (a+b+c) \cdot (a \times b) = c \cdot (a \times b) \]

2. **Second Term**: \( (a+b+c) \cdot (c \times a) \)

Similarly, \( a \cdot (c \times a) = 0 \) and \( b \cdot (c \times a) = 0 \). Thus:

\[ (a+b+c) \cdot (c \times a) = c \cdot (c \times a) = 0 \]

3. **Third Term**: \( (a+b+c) \cdot (c \times b) \)

Again, \( a \cdot (c \times b) = 0 \) and \( b \cdot (c \times b) = 0 \). Therefore:

\[ (a+b+c) \cdot (c \times b) = c \cdot (c \times b) = 0 \]

### Step 5: Combine Results
Combining all these results, we find:

\[ (a+b+c) \cdot (a \times b + c \times a + c \times b) = c \cdot (a \times b) + 0 + 0 = c \cdot (a \times b) \]

### Step 6: Final Expression
Now, we can substitute this back into our original expression:

\[ \frac{c \cdot (a \times b)}{[a \, b \, c]} \]

### Step 7: Recognize the Scalar Triple Product
Recall that the scalar triple product \( [a \, b \, c] \) can also be expressed as \( a \cdot (b \times c) \). Therefore, we can conclude:

\[ \frac{c \cdot (a \times b)}{[a \, b \, c]} = 1 \]

Thus, the final answer is:

\[ \boxed{1} \]