Let the vertex B of triangle[ABC] be located outside a circle & the other two vertices lie on the circumfrence. if the sides AB & BC intersects the circle at points D & E such that AD = EC, then prove that angleABC is equal to half the differences of the angles subtended by the chords AC & DE at the centre.
In ΔAOD and ΔCOE,
OA = OC (Radii of the same circle)
OD = OE (Radii of the same circle)
AD = CE (Given)
∴ ΔAOD ≅ ΔCOE (By SSS congruence- criterion)
∠OAD = ∠OCE (By Cpct) ... (i)
∠ODA = ∠OEC (By Cpct) ... (ii)
Also,
∠OAD = ∠ODA (As OA = OD) ... (iii)
From (i), (ii), and (iii), we get,
∠OAD = ∠OCE = ∠ODA = ∠OEC
Let ∠OAD = ∠OCE = ∠ODA = ∠OEC = x
In Δ OAC,
OA = OC
∴ ∠OCA = ∠OAC (Let a)
In Δ ODE,
OD = OE
∠OED = ∠ODE (Let y)
ADEC is a cyclic quadrilateral.
∴ ∠CAD + ∠DEC = 180° (Opposite angles are supplementary)
x + a + x + y = 180°
2x + a + y = 180°
y = 180º − 2x − a ... (iv)
Also,
∠DOE = 180º − 2y
and
∠AOC = 180º − 2a
∠DOE − ∠AOC = 2a − 2y = 2a − 2 (180º − 2x − a)
= 4a + 4x − 360° ... (v)
∠BAC + ∠CAD = 180º (Linear pair)
⇒ ∠BAC = 180º − ∠CAD = 180º − (a + x)
Similarly, ∠ACB = 180º − (a + x)
In ΔABC,
∠ABC + ∠BAC + ∠ACB = 180º (Angle sum property of a triangle)
∠ABC = 180º − ∠BAC − ∠ACB
= 180º − (180º − a − x) − (180º − a −x)
= 2a + 2x − 180º