The Circle is a fundamental geometric shape and the most aesthetic of all curves as well as the most symmetric. The construction of a circle using a compass is a granted one from the third postulate of Euclid. Circle exhibits wonderful geometric properties that are far from being trivial and have been a source of curiosity and pleasure for mathematicians for ages. For eg, of all plane shapes with a fixed perimeter (boundary) circle bounds the greatest amount of area (surface). We will explore some of these properties in this article.
DEFINITIONS
Circle: A closed curve lying in a plane, and such that all of its points are equally distant from a fixed point in the plane, is called a circle. That fixed point is called the center.
Circle as a Locus: It follows that the locus of a point in a plane at a given distance from a fixed point is a circle.
Radius: A straight line from the center to the circle is called a radius.
Equal Radii: It follows that all radii of the same circle or of equal circles are equal, and that all circles of equal radii are equal.
Diameter: A straight line through the center, terminated at each end by the circle, is called a diameter. Since a diameter equals two radii, it follows that all diameters of the
same circle or of equal circles are equal.
Arc: Any portion of a circle is called an arc. An arc that is half of a circle is called a semicircle. An arc less than a semicircle is called a minor arc, and an arc greater then a semicircle is called a major arc. The word arc taken alone is generally understood to mean a minor arc.
Central Angle: If the vertex of an angle is at the center of a circle and the sides are radii of the circle, the angle is called a central angle. An angle is said to intercept any arc cut cff by its sides, and the arc is said to subtend the angle.
Circular Arc Theorems
- In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc.
- In the same circle or in equal circles equal arcs subtend equal central angles; and of two unequal arcs the greater subtends the greater central angle.
- In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords; and if two arcs are unequal, the greater is subtended by the greater chord. Corollary: In the same circle or in equal circles, the greater of two unequal major arcs is subtended by the lesser chord.
- In the same circle or in equal circles, if two chords are equal, they subtend equal arcs; and if two chords are unequal, the greater subtends the greater arc. Corollary: In the same circle or in equal circles the greater of two unequal chords subtends the less major arc.
- A line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it.
- A diameter bisects the circle.
- A line through the center that bisects a chord, not a diameter, is perpendicular to the chord.
- The perpendicular bisector of a chord passes through the center of the circle and bisects the arcs subtended by the chord.
- In the same circle or in equal circles equal chords are equidistant from the center, and chords equidistant from the center are equal.
- In the same circle or in equal circles, if two chords are unequal, they are unequally distant from the center, and the greater chord is at the less distance.
- In the same circle or in equal circles, if two chords are unequally distant from the center, they are unequal, and the chord at the less distance is the greater.
- A line perpendicular to a radius at its extremity on the circle is tangent to the circle.
- A tangent to a circle is perpendicular to the radius drawn to the point of contact.
- A perpendicular to a tangent at the point of contact passes through the center of the circle.
- A perpendicular from the center of a circle to a tangent passes through the point of contact.
- Two parallel lines intercept equal arcs on a circle.
- Through three points not in a straight line one circle, and only one, can be drawn.
- In the same circle or in equal circles two central angles have the same ratio as their intercepted arcs.
- An inscribed angle is measured by half the intercepted arc.
- An angle formed by two chords intersecting within the circle.is measured by half the sum of the intercepted arcs.
- An angle formed by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc.
- An angle formed by two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured by half the difference of the intercepted arcs.
- An angle inscribed in a semicircle is a right angle. For it is half of a central straight angle.
- An angle inscribed in a segment greater than a semicircle is an acute angle, and an angle inscribed in a segment less than a semicircle is an obtuse angle.
- Angles inscribed in the same segment or in equal segments are equal.
- If a quadrilateral is inscribed in a circle, the opposite angles are supplementary; and, conversely, if two opposite angles of a quadrilateral are supplementary, the quadrilateral can be inscribed in a circle.
- To find the center of a given circle.
- INVERSE POINTS
- Two inverse points divide the corresponding diameter harmonically.
- The ratio of the distances of a variable point of a circle from two given inverse points is constant.
- In two orthogonal circles the two radii passing through a common point of the two circles are rectangular. Conversely, If the two radii passing through a point common to two circles are rectangular, the two circles are orthogonal.
- If two circles are orthogonal, the radius of one circle passing through a point common to the two circles is tangent to the second circle. Conversely, given two intersecting circles, if the radius of one circle passing through a point common to the two circles is tangent to the second circle, the two circles are orthogonal.
- If two circles are orthogonal, any two points of one of them collinear with the center of the second circle are inverse for that second circle. Conversely, If two points of one circle are inverse for a second circle, the two circles are orthogonal.
- The two lines joining the points of intersection of two orthogonal circles to a point on one of the circles meet the other circle in two diametrically opposite points.