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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 1 of 5: Parameterize the angles
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In plain words

Use the angle bisector at B B .

x=12∠ABC,∠A=60∘,∠C=120∘−2xx=\frac12\angle ABC,\quad\angle A=60^\circ,\quad\angle C=120^\circ-2x
Detailed analysis

Set x=∠ABQ=∠QBC x=\angle ABQ=\angle QBC . Then ∠A=60∘\angle A=60^\circ and ∠C=120∘−2x\angle C=120^\circ-2x .