## The theorem of Hippocrates (470 – c. 410 BC)

A halfcircle on *AB* is constructed to pass through *C*,
as well as halfcircles on *BC* and *AC* as shown. The white
'crescent moons' on *BC* and *AC* are called LUNES.

1) Drag *A*, *B* or *C*.

2) What do you notice about the sum of the areas of the lunes? Make a conjecture.

(**Note**: The Greek mathematician, Hippocrates of Chios, is not to be
confused with the different Greek physician of the Hippocratic oath in
medicine.)

####

The theorem of Hippocrates (470 – c. 410 BC)

**Challenge**: Can you *explain why* (prove) your conjecture above is true?

**Further Extension & Variation**:

1) Hippocrates' Theorem generalizes to a 'right quadrilateral', namely, a quadrilateral with a pair of opposite right angles as shown in the first figure below. Here the sum of the lunes on the sides is equal to the area of the right quadrilateral^{1}.

2) The second figure shows a variation of the theorem and in this case, the lilac (purplish-grey) region minus the red region is equal to the area of triangle *ABC*. Can you explain why (prove) this is true?

......

**Notes**:

1) The term 'right quadrilateral' was also used in my 1994/2009 book *Some Adventures in Euclidean Geometry*, pp. 73-74 for a quadrilateral with a pair of opposite right angles.

2) Hippocrates' Theorem or variations of it appears on and off in Mathematics Competitions around the world. For example, see Q10 of the *2009 Senior 2nd Round* of the South African Mathematics Olympiad. The theorem also appeared in the *Daily Problem*, AMESA Congress, June 2009 and was used by Phadiela Cooper from COSAT to investigate the collaborative problem solving of children in small groups - the topic of her plenary at the AMESA Congress, University of the Western Cape, 24-28 June 2013.

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By Michael de Villiers. Created, 31
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