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Centroid of Triangle

In ΔABC, the mid-points of the sides BC, CA, and AB are D, E, and F respectively. The lines AD, BE, and CF are called medians of the triangle ABC, the points of concurrency of three medians is called centroid. Generally it is represented by G.
Also, AG = 70727.pngAD, BG = 70721.png BE, and CG = 70715.pngCF
Apollonius theorem
AB2 + AC2 = 2(AD2 + BD2)

Escribed circles of a triangle and their radii

The circle which touches the side BC and two sides AB and AC produced of a triangle ABC is called the escribed circle opposite to the angle A. Its radius is denoted by r1. Similarly, r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively. The centers of the escribed circles are called the ex-centers. The center of the escribed circle opposite to the angle A is the point of intersection of the external bisectors of angles B and C. The internal bisector of angle A also passes through the same point. The center is generally denoted by I1.
In a ΔABC, we have
  1. r1 = 70432.png
  2. r1 = s tan 70426.png
  3. r1 = 4R sin 70419.png cos 70413.png cos 70407.png,
    r2 = 4R cos 70401.png sin 70395.pngcos 70389.png,
    r3 = 4R cos 70383.png
m – n Theorem
Let D be a point on the side BC of a ΔABC such that BDDC = m : n and ADC = θ, BAD = α, and DAC = β. Then
  1. (m + n) cot θ = m cot αn not β
  2. (m + n) cot θ = n cot Bm cot C

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