Question

Let the vertexof an angle ABC bd located out side s circle and let thesides of the angle intersect equal chords AD and CE with the circle . Prove that angle ABC in half the difference of the angles subtened by the chords AD ans CE

Answer 1

see the diagram.

ΔAOD and ΔCOE are Isosceles and Congruent.  Reasons:
     AD = CE  given,      AO = OD = OC = OE = Radius.
So  ∠DAO = ∠ADO = ∠ECO = ∠CEO = a (say)

Let ∠AOC = x,   ∠DOE = y   and  ∠ABC = z.    To prove that 2 z = | x - y |

At O,  ∠x + ∠y  = 2π - ∠AOD - ∠COE
                        = 2π - 2 (π - 2a)
           ∠x + ∠y = 4 a    ----- (1)

In the Isosceles ΔODE,   ∠ODE = ∠OED = (π-y)/2

At point D or at E,    ∠BED =∠BDE = π - a - (π-y)/2 = π/2 - a + y/2

In ΔBDE,  z = π - 2 * [π/2 - a + y/2 ]
         or,      ∠ABC = 2 a - y = (x+y)/2 - y/2
                              = (x - y)/2