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 2 of 5: Apply the sine rule
Previous stepNext step
In plain words

Compute the segments in the two triangles.

BP=ABsin⁡30∘sin⁡(150∘−2x),AQ=ABsin⁡xsin⁡(120∘−x),QB=ABsin⁡60∘sin⁡(120∘−x)BP=AB\frac{\sin30^\circ}{\sin(150^\circ-2x)},\quad AQ=AB\frac{\sin x}{\sin(120^\circ-x)},\quad QB=AB\frac{\sin60^\circ}{\sin(120^\circ-x)}
Detailed analysis

The sine rule in ABP ABP and ABQ ABQ gives these three formulas.