If `p_(1),p_(2),p_(3)` are altitudes of a triangle ABC from the vertices A,B,C and`!` the area of the triangle, then `p_(1).p_(2),p_(3)` is equal to

Options

A. abc
B. 8R
C. `a^(2)b^(2)c^(2)`
D. `(a^(2)*b^(2)*c^(2))/(8R^(3))`

Answer

To find the product of the altitudes \( p_1, p_2, p_3 \) of triangle \( ABC \) in terms of the area \( ! \) of the triangle and the circumradius \( R \), we can follow these steps:

### Step 1: Write the area of the triangle in terms of altitudes
The area \( ! \) of triangle \( ABC \) can be expressed using each altitude:
\[ ! = \frac{1}{2} \times A \times p_1 = \frac{1}{2} \times B \times p_2 = \frac{1}{2} \times C \times p_3 \]
where \( A, B, C \) are the lengths of the sides opposite to vertices \( A, B, C \) respectively.

### Step 2: Express the altitudes in terms of the area
From the area formulas, we can express each altitude:
\[ p_1 = \frac{2!}{A}, \quad p_2 = \frac{2!}{B}, \quad p_3 = \frac{2!}{C} \]

### Step 3: Find the product of the altitudes
Now, we can find the product \( p_1 \times p_2 \times p_3 \):
\[ p_1 \times p_2 \times p_3 = \left(\frac{2!}{A}\right) \times \left(\frac{2!}{B}\right) \times \left(\frac{2!}{C}\right) = \frac{(2!)^3}{A \times B \times C} \]
This simplifies to:
\[ p_1 \times p_2 \times p_3 = \frac{8!^3}{A \times B \times C} \]

### Step 4: Use the relationship between area, sides, and circumradius
We know from triangle properties that:
\[ ! = \frac{A \times B \times C}{4R} \]
Substituting this into our expression for \( p_1 \times p_2 \times p_3 \):
\[ p_1 \times p_2 \times p_3 = \frac{8 \left(\frac{A \times B \times C}{4R}\right)^3}{A \times B \times C} \]

### Step 5: Simplify the expression
This simplifies to:
\[ p_1 \times p_2 \times p_3 = \frac{8 \cdot A^2 \cdot B^2 \cdot C^2}{64R^3} = \frac{A^2 \cdot B^2 \cdot C^2}{8R^3} \]

### Final Result
Thus, the product of the altitudes \( p_1, p_2, p_3 \) is given by:
\[ p_1 \times p_2 \times p_3 = \frac{A^2 \cdot B^2 \cdot C^2}{8R^3} \]