Question

TA is a tangent to the circle from a point T and TBC is a secant to the circle.
If AD is the bisector of angle CAB, prove that ADT is isosceles.​

Answer 1

ΔADT is isosceles  if TA is a tangent to the circle from a point T and TBC is a secant to the circle and AD is the bisector of ∠CAB

Step-by-step explanation:

∠TAB = ∠BCA  =   P  (Alternate segment theorem)

∠CAD = ∠BAD = Q  ( as AD bisects ∠CAB)

∠TAD  = ∠TAB + ∠BAD

=> ∠TAD = P + Q

∠ADT  = ∠DCA  + ∠CAD

=> ∠ADT  = ∠BCA  + ∠CAD

=> ∠ADT  = P  +  Q

=> ∠TAD =   ∠ADT

in Δ ADT

∠TAD =   ∠ADT

=> ΔADT is isosceles

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