Let AD be a median of the `Delta ABC`. If AE and AF are medians of the triangle ABD and ADC, respectively, and BD=a/2 AD = `m_(1), AE = m_(2), AF = m_(3), " then " a^(2)//8` is equal to

Options

A. `m_(2)^(2) + m_(3)^(2) - 2m_(1)^(2)`
B. `m_(1)^(2) + m_(2)^(2) - 2m_(3)^(2)`
C. `m_(1)^(2) + m_(3)^(2) - 2m_(2)^(2)`
D. none of these

Explanation

To solve the problem, we will use the Apollonius Theorem, which relates the lengths of the sides of a triangle to the length of the median. Let's break down the steps:

### Step 1: Understand the given information
We have triangle ABC with median AD. BD is given as \( \frac{a}{2} \), where \( a \) is the length of side BC. We denote:
- \( AD = m_1 \)
- \( AE = m_2 \) (median of triangle ABD)
- \( AF = m_3 \) (median of triangle ADC)

### Step 2: Apply the Apollonius Theorem to triangle ABC
According to the Apollonius Theorem:
\[ AB^2 + AC^2 = 2AD^2 + 2BD^2 \]
Substituting the known values:
\[ AB^2 + AC^2 = 2m_1^2 + 2\left(\frac{a}{2}\right)^2 \]
Calculating \( BD^2 \):
\[ BD^2 = \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]
Thus, we have:
\[ AB^2 + AC^2 = 2m_1^2 + 2 \cdot \frac{a^2}{4} = 2m_1^2 + \frac{a^2}{2} \]
This is our **Equation (1)**.

### Step 3: Apply the Apollonius Theorem to triangle ABD
For triangle ABD, we can write:
\[ AB^2 + AD^2 = 2AE^2 + 2BD^2 \]
Substituting the known values:
\[ AB^2 + m_1^2 = 2m_2^2 + 2\left(\frac{a}{2}\right)^2 \]
This gives us:
\[ AB^2 + m_1^2 = 2m_2^2 + \frac{a^2}{4} \]
This is our **Equation (2)**.

### Step 4: Apply the Apollonius Theorem to triangle ACD
For triangle ACD, we have:
\[ AC^2 + AD^2 = 2AF^2 + 2CD^2 \]
Since \( CD = BD = \frac{a}{2} \):
\[ AC^2 + m_1^2 = 2m_3^2 + 2\left(\frac{a}{2}\right)^2 \]
This gives us:
\[ AC^2 + m_1^2 = 2m_3^2 + \frac{a^2}{4} \]
This is our **Equation (3)**.

### Step 5: Combine Equations (2) and (3)
Now we add Equations (2) and (3):
\[ (AB^2 + m_1^2) + (AC^2 + m_1^2) = (2m_2^2 + \frac{a^2}{4}) + (2m_3^2 + \frac{a^2}{4}) \]
This simplifies to:
\[ AB^2 + AC^2 + 2m_1^2 = 2m_2^2 + 2m_3^2 + \frac{a^2}{2} \]
Now, substituting from Equation (1):
\[ 2m_1^2 + \frac{a^2}{2} = 2m_2^2 + 2m_3^2 + \frac{a^2}{2} \]
Cancelling \( \frac{a^2}{2} \) from both sides, we get:
\[ 2m_1^2 = 2m_2^2 + 2m_3^2 \]
Dividing by 2:
\[ m_1^2 = m_2^2 + m_3^2 \]

### Step 6: Final result
From the above, we can express:
\[ \frac{a^2}{8} = m_2^2 + m_3^2 - 2m_1^2 \]

### Conclusion
Thus, we conclude that:
\[ \frac{a^2}{8} = m_2^2 + m_3^2 - 2m_1^2 \]