Question

If a + b + c = 2s, then prove the following identities

(a) s² + (s − a)² + (s − b)² + (s − c)² = a² + b² + c²

(b) a² + b² − c² + 2ab = 4s (s − c)

(c) c² + a² − b² + 2ca = 4s (s − b)

(d) a² − b² − c² + 2ab = 4(s − b) (s − c)

(e) (2bc + a² − b² − c²) (2bc − a² + b² + c²) = 16s (s − a) (s − b) (s − c)

(f)

Answer 1

It is given that a + b + c = 2s

(a) We need to show that

Consider the left hand side.

Hence, .

(b) We need to show that

Consider the left hand side.

Hence,

(c) We need to show that

Consider the left hand side.

Hence,

(d) We need to show that

Consider the left hand side.

Using, a + b + c = 2s, we get

Hence,

(e) We need to show that

Consider the left hand side.

Using a + b + c = 2s, we get:

Hence,

(f) We need to show that

Consider the left hand side.

Using, a + b + c = 2s, we get

Hence,