If a4+b4+c4 = 2c2(a2+b2), prove that C = 45o or 135o.

If a⁴+b⁴+c⁴= 2c²(a²+b²), prove that C = 45^o or 135^o.

1 Answers

a⁴ + b⁴ +c⁴= 2c² (a²+b²)

or, (a²+b²)²- 2a²b² + c⁴ = 2(a²+b²)c² [a²+ b² = (a+b)² - 2ab]

or, (a²+b²)²- 2(a²+b²)c² + (c2)2 = 2a²b²

or, (a²+ b²- c2)2 = 2a²b²

or, a²+ b²- c2= ±√2 ab

or, $\frac{a^{2}+ b^{2} - c^{2}}{ab}$ = ±√2

or, $\frac{a^{2}+ b^{2} - c^{2}}{2ab}$ = ± $\frac{\sqrt{2}}{2}$

or, cos C = $\pm \frac{1}{\sqrt{2}}$

or, cos C = cos 45° or cos 135°

or, C = 45° or 135°