MathLabs

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 3 of 5: Substitute the length condition
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In plain words

Cancel AB AB .

1+sin⁡30∘sin⁡(150∘−2x)=sin⁡x+sin⁡60∘sin⁡(120∘−x)1+\frac{\sin30^\circ}{\sin(150^\circ-2x)}=\frac{\sin x+\sin60^\circ}{\sin(120^\circ-x)}
Detailed analysis

Substitute into AB+BP=AQ+QB AB+BP=AQ+QB and divide by AB AB .