Solved: "If 8R2=a2+b2+c2 ,prove that the triangle is right..."

Nannie Mack

Answered question

2021-01-27

8 R^{2} = a^{2} + b^{2} + c^{2}

,prove that the triangle is right angle triangle. Here a,b and c are the lengths of side of triangle and R is circum radius.

irwchh

Skilled2021-01-28Added 102 answers

a = 2 r \sin A

b = 2 r \sin B

c = 2 r \sin C

a^{2} + b^{2} + c^{2} = 4 R^{2} (\sin^{2} A + \sin^{2} B + \sin^{2} C) = 8 R^{2}

\sin^{2} A + \sin^{2} B + \sin^{2} C = 2

1 - \cos 2 A + 1 - \cos 2 B + 1 - \cos 2 C = 4

\cos 2 A + \cos 2 B + \cos 2 C + 1 = 0

\cos A = - \cos (B + C)

1 + \cos 2 A = 2 \cos (B + C) \cos (B + C)

2 \cos (B + C) \cos (B + C) + \cos 2 B + \cos 2 C = 2 \cos (B + C) \cos (B + C) + 2 \cos (B + C) \cos (B - C)

2 \cos (B + C) (\cos (B + C) + \cos (B - C)) = 0

4 \cos A \cos B \cos C = 0

Some of the factors must be zero so one angle is right

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