SOLUTION: In triangle ABC, ray l bisects angle B, and intersects side AC in point D. Prove that AB/BC = AD/DC.

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Question 429744: In triangle ABC, ray l bisects angle B, and intersects side AC in point D. Prove that AB/BC = AD/DC.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!

Applying the law of sines on triangles ABD and DBC,
AD%2Fsin%28theta%29+=+AB%2Fsin%28alpha%29
DC%2Fsin%28theta%29+=+BC%2Fsin%28pi+-+alpha%29+=+BC%2Fsin%28alpha%29
The first equation is equivalent to ADsin%28alpha%29+=+ABsin%28theta%29 --> sin%28alpha%29%2Fsin%28theta%29+=+AB%2FAD. Similarly, the second equation is equivalent to DCsin%28alpha%29+=+BCsin%28theta%29 --> sin%28alpha%29%2Fsin%28theta%29+=+BC%2FDC. Hence, we have sin%28alpha%29%2Fsin%28theta%29+=+AB%2FAD+=+BC%2FDC. Cross-multiplying, AB%2ADC+=+AD%2ABC. Multiplying both sides by 1%2F%28BC%2ADC%29 we have AB%2FBC+=+AD%2FDC.