circle question(with picture)

圖片參考：http://eclass.twgss.edu.hk/inabox/files/c219/question/9037879b95c3211d756f0caa27d7369c_1126525267ecPackage/ecPackage/20050912_194107/Circle/angle/1987Q45.jpg

Let d be the remaining angle
Then a + b + c + d = 360°.............(1)
Consider the reflex angle 360° - c. This is an angle at the centre. The corresponding angle at the circumference is d. Thus 360° - c = 2d..........(2)
From (1), a + b = 360° - c - d
= 360° - c - (360° - c) / 2............[from (2)]
= (360° - c) / 2
= 180° - (c / 2)

Let the remaining angle be D,
D = (360-c)/2 ---------( angle at centre twice angle at circum.)
a+b+c+d=360--------------(angles at a point)
a+b+c+(360-c)/2=360
c=2(180-a-b)

You have a quadrilateral inside a circle, with angles A, B and C, let the last one be D and the inverse of that (ie 360 - D) be E.

Now E = 2C (law of circle - angle @ circumference is 2X the angle @ center)
and D = 360 - (A+B+C) (law of quadrilateral - sum of all angles in quad. = 360)

Since E + D = 360

so
360 = 2C + (360 - (A+B+C))
360 = 360 - A - B + C
0 = - A - B + C
A+B = C