Lune of Hippocrates - Proof

Proof

Hippocrates' result can be proved as follows: The center of the circle on which the arc AEB lies is the point D, which is the midpoint of the hypotenuse of the isosceles right triangle ABO. Therefore the diameter AC of the larger circle ABC is √2 times the diameter of the smaller circle on which the arc AEB lies. Consequently the smaller circle has half the area of the larger circle, and therefore the quarter circle AFBOA is equal in area to the semicircle AEBDA. Subtracting the crescent-shaped area AFBDA from the quarter circle gives triangle ABO and subtracting the same crescent from the semicircle gives the lune. Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area.

Read more about this topic:  Lune Of Hippocrates

Famous quotes containing the word proof:

    To cease to admire is a proof of deterioration.
    Charles Horton Cooley (1864–1929)

    Right and proof are two crutches for everything bent and crooked that limps along.
    Franz Grillparzer (1791–1872)

    Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?
    Henry David Thoreau (1817–1862)