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Delta ABC is inscribed in a circle with...

`Delta ABC ` is inscribed in a circle with centre O and line AP is a tangent at the point A. Ray AF is the bisector of `/_ BAC `
Prove that seg `AP ~= ` seg PE.

Text Solution

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Line AP is a tangent and seg AC is a chord.
`:. /_ PAC = /_ ABP ` …..(Angles in alternate segmets ) …(1)
`/_ PAE = /_ PAC + /_CAE ` …(Angle addition postulate) …(2)
Ray AF is the bisector of `/_BAC` …..(Given ) ….(3)
From (1), (2) and (3) ,
`/_ PAE = /_ ABP + (1)/(2) /_BAC` ....(4)
`/_ AEP = /_ BAE + / _ ABP` ...(Ray AF is the bisector of `/_ BAC ) ` ...(6)
From ( 4) and (6) ,
`/_PAE = /_ AEP `
`:. `seg PA = seg PE ...(Sides opposite to equal angles in a triangle )
i.e., seg PA `~= ` seg PE.
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