Question

If \( H \) is the orthocenter of \( \triangle ABC \) and \( AH = x \), \( BH = y \), \( CH = z \), then evaluate: \[ \frac{abc}{xyz} \]

• A. \( 1 \)
• B. \( \frac{a+b+c}{x+y+z} \)
• C. \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} \)
• D. \( \frac{ab + bc + ca}{xy + yz + zx} \)

Hint:

For an orthocenter \( H \) in \( \triangle ABC \), the relation \( \frac{abc}{xyz} = \frac{a}{x} + \frac{b}{y} + \frac{c}{z} \) holds.

Answer 1

The Correct Option is C

Using the standard result from triangle geometry: \[ \frac{abc}{xyz} = \frac{a}{x} + \frac{b}{y} + \frac{c}{z} \] Thus, the correct answer is \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} \).