If `p_(1), p _(2), p_(3)` are respectively the perpendicular from the vertices of a triangle to the opposite sides, then find the value of `p_(1) p_(2)p _(3).`

Options

A. `(a^(2)b^(2)c^(2))/(R^(2))`
B. `(a^(2)b^(2)c^(2))/(4R^(2))`
C. `(4a^(2)b^(2)c^(3))/(R^(2))`
D. `(a^(2)b^(2)c^(2))/(8R^(2))`

Answer

To find the value of \( p_1 p_2 p_3 \), where \( p_1, p_2, p_3 \) are the perpendiculars from the vertices of a triangle to the opposite sides, we can follow these steps:

### Step 1: Understand the Triangle and Define the Perpendiculars
Let \( \triangle ABC \) be a triangle with sides \( a, b, c \) opposite to vertices \( A, B, C \) respectively. The perpendiculars from vertices \( A, B, C \) to the opposite sides are denoted as \( p_1, p_2, p_3 \).

### Step 2: Express the Area of the Triangle
The area \( \Delta \) of triangle \( ABC \) can be expressed in terms of the base and height. For each vertex, we can express the area as:
- From vertex \( B \):
\[ \Delta = \frac{1}{2} \times b \times p_1 \implies p_1 = \frac{2\Delta}{b} \]
- From vertex \( C \):
\[ \Delta = \frac{1}{2} \times c \times p_2 \implies p_2 = \frac{2\Delta}{c} \]
- From vertex \( A \):
\[ \Delta = \frac{1}{2} \times a \times p_3 \implies p_3 = \frac{2\Delta}{a} \]

### Step 3: Multiply the Expressions for \( p_1, p_2, p_3 \)
Now, we can multiply these three equations:
\[ p_1 p_2 p_3 = \left(\frac{2\Delta}{b}\right) \left(\frac{2\Delta}{c}\right) \left(\frac{2\Delta}{a}\right) \]
This simplifies to:
\[ p_1 p_2 p_3 = \frac{8\Delta^3}{abc} \]

### Step 4: Relate Area to Circumradius
We know from triangle properties that the area \( \Delta \) can also be expressed in terms of the circumradius \( R \):
\[ \Delta = \frac{abc}{4R} \]
Substituting this into the equation for \( p_1 p_2 p_3 \):
\[ p_1 p_2 p_3 = \frac{8\left(\frac{abc}{4R}\right)^3}{abc} \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
\[ p_1 p_2 p_3 = \frac{8 \cdot \frac{a^3b^3c^3}{64R^3}}{abc} = \frac{8a^2b^2c^2}{64R^3} = \frac{a^2b^2c^2}{8R^3} \]

### Final Result
Thus, the value of \( p_1 p_2 p_3 \) is:
\[ p_1 p_2 p_3 = \frac{a^2b^2c^2}{8R^3} \]