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Problem 5

In triangle ABC ABC , AP AP bisects ∠BAC\angle BAC with P P on BC BC , and BQ BQ bisects ∠ABC\angle ABC with Q Q on CA CA . Given ∠BAC=60∘\angle BAC=60^\circ and AB+BP=AQ+QB AB+BP=AQ+QB , determine the possible angles of ABC ABC .
Step 5 of 5: Select the admissible root
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In plain words

Exclude the degenerate case.

∠ABC=80∘\angle ABC=80^\circ
Detailed analysis

The other factor gives the degenerate ∠B=120∘\angle B=120^\circ ; the admissible root is t=20∘ t=20^\circ , so ∠B=80∘\angle B=80^\circ and ∠C=40∘\angle C=40^\circ .