QQuestion
If \frac{x}{y+z} = \frac{y}{z+x} = \frac{z}{ x+y}, then show that each ratio is \frac{1}{ 2} or -1.
✓Answer
\frac{x}{y+z} = \frac{y}{z+x} = \frac{z}{x+y} = k (say)
⇒ x = k(y+z) — (i)
⇒ y = k(z+x) — (ii)
⇒ z = k(x+y) — (ii)
(i) + (ii) + (iii)
x + y + z = k(y+z) + k(z+x) + k(x+y)
or, x + y + z = k(y + z + z + x + x + y)
or, x + y + z = 2k(x + y + z)
or, (x + y + z) – 2k(x + y + z) = 0
or, (x + y + z) (1 – 2k) = 0
either, x + y + z = 0 or, 1 – 2k = 0 ⇒ k = \frac{1}{2}
Again,
| x + y + z = 0
⇒ y + z = – x ⇒ \frac{x}{y + z} = -1 |
x + y + z = 0
⇒ x + z = – y ⇒ \frac{y}{x + z} = -1 |
x + y + z = 0
⇒ x + y = – z ⇒ \frac{z}{x + y} = -1 |
\frac{x}{y + z} = \frac{y}{x + z} = \frac{z}{x + y} = -1 or \frac{1}{2} (Proved)
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