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TA is tangent to the circle at point A, TBC is the secant intersecting the circle A,B and C . AD is the bisector of ,<CAB.
<TAB=<BCA (alternate segment theorem)____________ (1)
AD is bisector of <BAC=<BAD + <DAC____ (2)
<TAD=<TAB + <BAD
= <BCA + <DAC (from (1) and(2)
=<DCA + <DAC
=180 - < CDA (in triangle CAD,<CDA + <DCA + <ACD = 180 )
= <TAD= <TDA (<CDA +<TDA=180)
= TD = TA (side oppo. to equal angle are equal)
therefore,triangle ADT is an isosceles triangle.
<TAB=<BCA (alternate segment theorem)____________ (1)
AD is bisector of <BAC=<BAD + <DAC____ (2)
<TAD=<TAB + <BAD
= <BCA + <DAC (from (1) and(2)
=<DCA + <DAC
=180 - < CDA (in triangle CAD,<CDA + <DCA + <ACD = 180 )
= <TAD= <TDA (<CDA +<TDA=180)
= TD = TA (side oppo. to equal angle are equal)
therefore,triangle ADT is an isosceles triangle.
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