Euclidean geometry Circles

A chord AB is a segment in the interior of a circle connecting two points (A and B) on the circumference. When a chord passes through the circle’s centre, it is a diameter, d. The circumference of a circle is given by πd, or 2πr where r is the radius of the circle; the area of a circle is πr². In each case, π is the same constant (3.14159…). The Greek mathematician Archimedes (c. 285–212/211 bc) used the method of exhaustion to obtain upper and lower bounds for π by circumscribing and inscribing regular polygons about a circle (see animation).

A semicircle has its end points on a diameter of a circle. Thales (flourished 6th century bc) is generally credited with proving that any angle inscribed in a semicircle is a right angle; that is, for any point C on the semicircle with diameter AB, ∠ACB will always be 90 degrees (see Sidebar: Thales’ Rectangle). Another important theorem states that for any chord AB in a circle, the angle subtended by any point on the same semiarc of the circle will be invariant (see figure). Slightly modified, this means that in a circle, equal chords determine equal angles, and vice versa.

Summarizing the above material, the five most important theorems of plane Euclidean geometry are: the sum of the angles in a triangle is 180 degrees, the Bridge of Asses, the fundamental theorem of similarity, the Pythagorean theorem, and the invariance of angles subtended by a chord in a circle. Most of the more advanced theorems of plane Euclidean geometry are proven with the help of these theorems.