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In any triangle Sin A / Sin B = BC / AC

Ratio of Sine of angles = Ratio of sides opposite the angles.

Sin ABD / sin ADB  = AD / AB

Sin DBC / Sin BDC = DC / BC

Angle bisector : =>  angle ABD = angle DBC
and sin ADB = sin (180 - BDC) = sin BDC

Hence we have :  AD / AB = DC / BC

THis is the angle bisector theorem.