| Manifesto | Bookstore | Contents | Amazon store | Term index | What changed? | Contact | Recommend |
Why a Diameter Is the Longest Chord?Indeed why? Why a diameter is the longest chord in a circle? I sometimes heard and several times read at popular math sites that the reason why a diameter is the longest chord is that a diameter passes through the center of the circle. While that is true that passing through the center has something to do with the length of a chord, the answer, as given, is vacuous. Since the definition of a diameter is a chord passing through the center of the circle, such an explanation actually reads: "The diameter is the longest chord in a circle because it is a diameter of the circle." How much does that explain? Let's recollect the definitions:
As the statement is definitely not an axiom of geometry, it must be proved, i.e., logically derived from simpler statements and the definitions. Incidently, Euclid proved that statement in the third book of his Elements as a more informative Proposition XV:
On a casual inspection, it seems obvious:
Nonetheless, it has to be proved and Euclid proves the first part with the reference to the Triangle Inequality (I.20) and the second part to the Pythagorean theorem.
Let AB be a diameter of the circle with center C and DE a chord not through C. Then, by the definition of the circle as the locus of points equidistant from the center, On the other hand, in ΔCDE, by the triangle inequality,
The same route is taken in my favorite Kiselev's Geometry. In another geometric classics, Lessons in Geometry by J. Hadamard (whose new edition is planned for December, 2008) the triangle inequality is applied differently, after establishing the fact that, for a point P on the diameter AB, one of the ends of the diameter is farthest from P among all points of the circle and the other is nearest. References
|
| 34384457 |  |
