Difference between revisions of "Circle"
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===Olympiad===  ===Olympiad===  
+  *Consider a [[circle]] <math>S</math>, and a [[point]] <math>P</math> outside it. The [[tangent line]]s from <math>P</math> meet <math>S</math> at <math>A</math> and <math>B</math>, respectively. Let <math>M</math> be the [[midpoint]] of <math>AB</math>. The [[perpendicular bisector]] of <math>AM</math> meets <math>S</math> in a point <math>C</math> lying inside the [[triangle]] <math>ABP</math>. <math>AC</math> intersects <math>PM</math> at <math>G</math>, and <math>PM</math> meets <math>S</math> in a point <math>D</math> lying outside the triangle <math>ABP</math>. If <math>BD</math> is [[parallel]] to <math>AC</math>, show that <math>G</math> is the [[centroid]] of the triangle <math>ABP</math>.  
+  (<url>viewtopic.php?=217167 Source</url>)  
== See Also ==  == See Also == 
Revision as of 13:31, 30 March 2010
A circle is a geometric figure commonly used in Euclidean geometry.

A basic circle. 
Contents
Traditional Definition
A circle is defined as the set (or locus) of points in a plane with an equal distance from a fixed point. The fixed point is called the center and the distance from the center to a point on the circle is called the radius.
Coordinate Definition
Using the traditional definition of a circle, we can find the general form of the equation of a circle on the coordinate plane given its radius, , and center . We know that each point, , on the circle which we want to identify is a distance from . Using the distance formula, this gives which is more commonly written as
Example: The equation represents the circle with center and radius 5 units.
Area of a Circle
The area of a circle is where is the mathematical constant pi and is the radius.
Archimedes' Proof
We shall explore two of the Greek mathematician Archimedes demonstrations of the area of a circle. The first is much more intuitive.
Archimedes envisioned cutting a circle up into many little wedges (think of slices of pizza). Then these wedges were placed side by side as shown below:
As these slices are made infinitely thin, the little green arcs in the diagram will become the blue line and the figure will approach the shape of a rectangle with length and width thus making its area .
Archimedes also came up with a brilliant proof of the area of a circle by using the proof technique of reductio ad absurdum.
Archimedes' actual claim was that a circle with radius and circumference had an area equivalent to the area of a right triangle with base and height . First let the area of the circle be and the area of the triangle be . We have three cases then.
Case 1: The circle's area is greater than the triangle's area.
Related Formulae
 The area of a circle with radius is .
 The circumference of a circle with radius is .
Other Properties and Definitions

A circle with a tangent and a chord marked. 
 A line that touches a circle at only one point is called the tangent of that circle. Note that any point on a circle can have only one tangent.
 A line segment that has endpoints on the circle is called the chord of the circle. If the chord is extended to a line, that line is called a secant of the circle.
 Chords, secants, and tangents have the following properties:
 The perpendicular bisector of a chord is always a diameter of the circle.
 The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
 The Power of a point theorem.
Other interesting properties are:
 A right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
 Any angle formed by the two endpoints of a diameter of the circle and a third distinct point on the circle as the vertex is a right angle.
Problems
Introductory
Intermediate
 Circles with centers and have radii 3 and 8, respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and . What is ?
(Source)
 Let
 and
. What is the ratio of the area of to the area of ?
(Source)
Olympiad
 Consider a circle , and a point outside it. The tangent lines from meet at and , respectively. Let be the midpoint of . The perpendicular bisector of meets in a point lying inside the triangle . intersects at , and meets in a point lying outside the triangle . If is parallel to , show that is the centroid of the triangle .
(<url>viewtopic.php?=217167 Source</url>)