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Middle/High School Algebra, Geometry, and Statistics (AGS)
7.21 Circles and Concurrent Lines in Triangles - Review

  • The set of points in a plane which are a fixed distance from a fixed point is called a circle. The fixed point is the centre and the fixed distance is the radius of the circle.
  • If 'r' is the radius and 'd' is the diameter of a circle.
    Circumference = 2 * pi * r = pi * d
    Area of the circle = pi * r2
  • The chord passing through the centre of a circle is called a diameter. The diameter is the longest chord of the circle.
  • The perpendicular bisector of a chord passes through its centre. The perpendicular drawn from the centre of a circle to a chord bisects it. There exits one and only one circle through three non-collinear points.
  • If two chords of a circle are equal, then they are equidistant from the centre. Conversely chords equidistant form the centre are equal.
  • The medians of a triangle are concurrent. The point of concurrence is called the centroid and it is denoted by G. G divides each median in the ration 2:1.
  • The perpendicular bisector of the sides of a triangle are concurrent. The point is called circumcentre. It is equidistant from the vertices and this distance is called the circum radius. The circumcentre is denoted by S and the circum radius is denoted by R. The circle with centre S and passing through the vertices of the triangle is called circum circle.
  • The internal bisectors of the angles of a triangle are concurrent and the point of concurrency is called the incentre which is denoted by 'I'. The incentre is equidistant from the sides and this distance is called inradius. The circle with centre 'I' and radius equal to inradius is called the incircle which touches all the sides. 'I' is always in the interior of the triangle. The inradius is denoted by 'r'.
  • The altitudes of a triangle are concurrent and the point of concurrency is called the orthocenter which is denoted by 'O'.
    • The orthocenter of an acute triangle lies inside the triangle.
    • The orthocenter of a right angle is the vertex containing the right angle.
    • The orthocenter of an obtuse triangle lies in the exterior of the triangle.
  • The circumcentre, incentre, orthocenter and centroid of an isosceles triangle are collinear and for an equilateral triangle they are coincide.

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