07 Oct
Broken Chord to the Rescue
In mathematics, even the most obvious properties need to be proved, unless of course they are accepted as axioms. The following one seems pretty much obvious and still requires a proof:
P is a variable point on the arc of a circle cut off by the chord AB. Prove the intuitively obvious property that the sum of the chords AP and BP is a maximum when P is at the midpoint O of the arc AB.
Ross Honsberger included that problems in two of his books, Mathematical Morsels and Mathematical Gems III. The problem indeed admits several elegant proofs.
Here's a short one based on the Broken Chord Theorem.
Let OU be perpendicular to AP (assuming AP is longer than BP). By the Broken Chord Theorem,
AU = (AP + BP) / 2.
But AU is a leg in the right triangle AOU in which AO is the hypotenuse. It follows that
AO + BO = 2 AO > 2 AU = AP + BP.
Reference
- R. Honsberger, Mathematical Morsels, MAA, 1978
- R. Honsberger, More Mathematical Morsels, MAA, New Math Library, 1991
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