Expert: Ahmed Salami - 1/1/2010

Question
"Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference.Is it true that the area of triangle ABC is largest when the triangle is isosceles?How do you know?"

Answer
Hi Buse,
The described triangle is always a right-triangle at C. If x and y are the other two sides while the diameter d is the hypotenuse, then
x² + y² = d²
y = √(d² - x²)
The area of the triangle
A = xy/2
 = (x/2)√(d² - x²)
The maximum value of A occurs when dA/dx = 0
dA/dx = (x/2).[-2x / 2√(d² - x²)] + √(d² - x²) . (1/2)
     = [-x² / 2√(d² - x²)] + (1/2)√(d² - x²)
equating to zero,
[-x² / 2√(d² - x²)] + (1/2)√(d² - x²) = 0
-x²/2√(d² - x²) = -(1/2)√(d² - x²)
x² = √(d² - x²).√(d² - x²)
x² = d² - x²
x = √(d² - x²)
 = y
And we have thus showed that A is maximum when x = y, also notice that it doesnt even depend on the diameter of the circle.
I hope you understand it.

Regards.