If a, b, and c are the interior angles of triangle abc, prove that 4 sina sinb sinc = sin2a + sin2b + sin2c

NOTES:

1)⇒    A + B + C = 180°

       A + B + C = π

       A + B        = π - C

2)⇒⇒ sin (A + B) = sin (π - C)

                         = (sin π)(cos C) - (sin C)(cos π)

                          = (0)(cos C) - (sin C)(-1)

                          = 0 - (-sin C)

                          = sin C

3)⇒⇒⇒cos (A + B) = cos (π - c)

                            = (cos π)(cos C) + (sin π)(sin C)

                            = (-1)(cos C) + (0)(sin C)

                            = - cos C              

4)⇒⇒⇒⇒ sin 2A + sin 2B = 2 sin (A + B) cos (A - B)

PROOF (from left side):

sin 2A + sin 2B + sin 2C    =    4 sin A sin B sin C

2 sin (A + B) cos (A - B) + sin 2C     refer to NOTE 4

2 sin (A + B) cos (A - B) + 2 sin C cos C  double angle formula

2 sin C cos (A - B) + 2 sin C cos C   refer to NOTE 2

2 sin C [cos (A - B) + cos C]     factored out 2 sin C

2 sin C [cos (A - B) - (cos(A + B)]   refer to NOTE 3

2 sin C [2 sin A sin B]           sum/difference formula

4 sin A sin B sin C      multiplied 2 sin C by 2 sin A sin B

Proof completed: 4 sin A sin B sin C   =   4 sin A sin B sin C

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

add all the numbers and check the ones place.

the answer is 486. as the ones place is more than 4, the estimation will be the next 10 ie, 490