Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
General ideas

	(1)
	(2)

Prove that .

First proof | Second proof

First proof:

(1) + (2)

(3)

that is, 

Dividing by 2,

(4)

Squaring the both sides,

(5)

Claim

(6)

(1) - (2)

(7)

(8)

From , we have:
From equation (8), we have:

(9)

From (6), we have:

(10)

Squaring (9) and adding it to (10), we have:

From equations (5) and (6), we have:

Since ,

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From (1)		(3)
From (2)		(4)

where satisfies , or (if a = 0).

(3) + (4)
that is,

Since ,

Squaring the both sides,

(5)

Claim:

(6)

(3) - (4)  

Since ,

that is,

(7)

Since ,

so 

Since , .  Hence (6) holds. From equations (5) and (6) we have:

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