Archimedes' Quadruplets - Proof of Congruency

Proof of Congruency

According to Proposition 5 of Archimedes' Book of Lemmas, the common radius of Archimedes' twin circles is:

By the Pythagorean theorem:

Then, create two circles with centers Ji perpendicular to HE, tangent to the large semicircle at point Li, tangent to point E, and with equal radii x. Using the Pythagorean theorem:

Also:

Combinding these gives:

Expanding, collecting to one side, and factoring:

Solving for x:

Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas.

Read more about this topic:  Archimedes' Quadruplets

Famous quotes containing the words proof of and/or proof:

    “There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “The moment a man begins to talk about technique that’s proof that he is fresh out of ideas.”
    —Raymond Chandler (1888–1959)