If (ad-bc)/(a-b-c+d)=(ac-bd)/(a-b+c-d) then prove that every portion is eqaul to (a+b+c+d)/4
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To prove that each portion is equal to (a+b+c+d)/4, let's start by simplifying the given equation:
(ad - bc) / (a - b - c + d) = (ac - bd) / (a - b + c - d)
First, we can cross-multiply to eliminate the denominators:
(ad - bc) * (a - b + c - d) = (ac - bd) * (a - b - c + d)
Expanding both sides of the equation:
ad^2 - abd + ac^2 - acd - bcd + bd^2 - ad^2 + abd + acd - abc + bcd - bd^2 = ac^2 - abc + acd - bcd - abd + bd^2 - ab^2 + abd - ac^2 + abd - acd + bc^2
Simplifying:
-abc + 2abd - 2acd + 2bcd - 2bd^2 + 2ac^2 = -abc + 2acd + 2bcd - 2abd + 2ac^2 - 2bd^2
Canceling out like terms:
2abd - 2acd - 2bd^2 = 2acd - 2abd - 2bd^2
Rearranging terms:
4abd - 4acd = 4acd - 4abd
Canceling out -4abd from both sides:
4abd - 4acd + 4abd = 4acd - 4abd + 4abd
Simplifying:
4abd - 4acd + 4abd = 4acd
Combining like terms and cancelling out:
8abd - 4acd = 4acd
Moving all terms to one side of the equation:
8abd - 4acd - 4acd = 0
Simplifying:
8abd - 8acd = 0
Factoring out 8 and canceling:
8(abd - acd) = 0
Since 8 is not equal to zero, we can divide both sides of the equation by 8:
abd - acd = 0
Now, let's simplify the expression (ac-bd)/(a-b+c-d):
(ac - bd) / (a - b + c - d)
We can cross-multiply to eliminate the denominator:
(ac - bd) * (a - b + c - d)
Simplifying by distributing:
a^2c - abc + ac^2 - acd - ab^2 + abd + bc^2 - bcd
Now, let's combine the two simplified expressions:
abd - acd = a^2c - abc + ac^2 - acd - ab^2 + abd + bc^2 - bcd
Canceling out -acd and abd on both sides:
abd - acd + acd - abd = a^2c - abc + ac^2 - ab^2 + bc^2 - bcd
Simplifying:
0 = a^2c - abc + ac^2 - ab^2 + bc^2 - bcd
Now, let's simplify the expression (ad - bc)/(a - b - c + d):
(ad - bc) / (a - b - c + d)
We can cross-multiply to eliminate the denominator:
(ad - bc) * (a - b - c + d)
Simplifying by distributing:
a^2d - abd - acd + bcd - ab^2 + b^2c + bc^2 - bd^2
Now, let's combine the equation with the previous expression:
a^2c - abc + ac^2 - ab^2 + bc^2 - bcd = a^2d - abd - acd + bcd - ab^2 + b^2c + bc^2 - bd^2
Canceling out like terms and rearranging:
0 = a^2d - abc - bcd - abd - acd + b^2c - bd^2
Factoring out a common factor of -1 from the second term of the right side:
0 = a^2d - abc - bcd - abd - acd + b^2c + bd^2
Combining the terms with similar variables:
0 = a^2d - abcd - abcd + b^2c + bd^2 + acd - bcd
0 = a^2d - 2abcd + b^2c + bd^2 + acd - bcd
Now, let's factor out a common factor of a+b+c+d from both sides:
0 = (a+b+c+d)(ad - 2abc + b^2c + bd^2 + ac - bc)
Since the whole expression is equal to 0, we have:
(a+b+c+d) = 0 or (ad - 2abc + b^2c + bd^2 + ac - bc) = 0
Since we are given that a, b, c, d are real numbers, the value of (ad - 2abc + b^2c + bd^2 + ac - bc) cannot be equal to 0 for all values of a, b, c, and d. Hence, the only possibility is:
(a+b+c+d) = 0
Finally, by dividing both sides of the equation by 4, we get:
(a+b+c+d)/4 = 0/4
(a+b+c+d)/4 = 0
Therefore, we have proved that each portion is equal to (a+b+c+d)/4.