In an acute angled triangle `ABC`, let `AD, BE` and `CF` be the perpendicular opposite sides of the triangle. The ratio of the product of the side lengths of the triangles `DEF` and `ABC`, is equal to

Options

A. A. `(3(abc)^(1/3))/(4(a+b+c))`
B. B. `1/4`
C. C. `cos A cos B cos C`
D. D. `sin ((A)/(2))sin ((B)/(2))sin ((C)/(2))`

Answer

To solve the problem, we need to find the ratio of the product of the side lengths of triangle DEF to the product of the side lengths of triangle ABC. Let's go through the steps systematically:

### Step 1: Understand the Triangles
We have an acute-angled triangle ABC with perpendiculars AD, BE, and CF dropped from vertices A, B, and C to the opposite sides BC, CA, and AB respectively. The points D, E, and F are the feet of these perpendiculars, forming triangle DEF.

### Step 2: Identify the Sides of Triangle DEF
Using properties of the pedal triangle, we know the lengths of the sides of triangle DEF can be expressed in terms of the circumradius R and the angles of triangle ABC:
- \( DE = R \cdot \sin(2C) \)
- \( EF = R \cdot \sin(2A) \)
- \( DF = R \cdot \sin(2B) \)

### Step 3: Calculate the Product of the Sides of Triangle DEF
Now, we can calculate the product of the sides of triangle DEF:
\[ DE \cdot EF \cdot DF = (R \cdot \sin(2C)) \cdot (R \cdot \sin(2A)) \cdot (R \cdot \sin(2B)) = R^3 \cdot \sin(2A) \cdot \sin(2B) \cdot \sin(2C) \]

### Step 4: Identify the Sides of Triangle ABC
For triangle ABC, we denote the sides as:
- \( a = BC \)
- \( b = CA \)
- \( c = AB \)

Using the sine rule, we can express the sides in terms of the circumradius R:
- \( a = 2R \cdot \sin A \)
- \( b = 2R \cdot \sin B \)
- \( c = 2R \cdot \sin C \)

### Step 5: Calculate the Product of the Sides of Triangle ABC
Now we calculate the product of the sides of triangle ABC:
\[ a \cdot b \cdot c = (2R \cdot \sin A) \cdot (2R \cdot \sin B) \cdot (2R \cdot \sin C) = 8R^3 \cdot \sin A \cdot \sin B \cdot \sin C \]

### Step 6: Find the Ratio
Now we can find the ratio of the product of the sides of triangle DEF to the product of the sides of triangle ABC:
\[ \text{Ratio} = \frac{DE \cdot EF \cdot DF}{a \cdot b \cdot c} = \frac{R^3 \cdot \sin(2A) \cdot \sin(2B) \cdot \sin(2C)}{8R^3 \cdot \sin A \cdot \sin B \cdot \sin C} \]

### Step 7: Simplify the Ratio
We can simplify this expression:
\[ \text{Ratio} = \frac{\sin(2A) \cdot \sin(2B) \cdot \sin(2C)}{8 \cdot \sin A \cdot \sin B \cdot \sin C} \]
Using the identity \(\sin(2A) = 2 \sin A \cos A\), we can further simplify:
\[ \text{Ratio} = \frac{2 \sin A \cos A \cdot 2 \sin B \cos B \cdot 2 \sin C \cos C}{8 \sin A \sin B \sin C} \]
This simplifies to:
\[ \text{Ratio} = \cos A \cdot \cos B \cdot \cos C \]

### Final Answer
Thus, the ratio of the product of the side lengths of triangle DEF to the product of the side lengths of triangle ABC is:
\[ \cos A \cdot \cos B \cdot \cos C \]