Toolkit 108: Ratio Lemma — MathGoldMedalist

By the Law of Sines in triangles ABP and APC,

\frac{\sin A_1}{BP} = \frac{\sin B}{AP} = \frac{\sin P_1}{c}

\frac{\sin A_2}{PC} = \frac{\sin C}{AP} = \frac{\sin P_2}{b}

By dividing,

\frac{\dfrac{\sin A_1}{BP}} {\dfrac{\sin A_2}{PC}} = \frac{\dfrac{\sin B}{AP}} {\dfrac{\sin C}{AP}} = \frac{\dfrac{\sin P_1}{c}} {\dfrac{\sin P_2}{b}}

\frac{\sin A_1}{\sin A_2}\cdot\frac{PC}{BP} = \frac{\sin B}{\sin C} = \frac{b}{c}

\frac{\sin A_1}{\sin A_2} = \frac{BP}{PC}\cdot\frac{\sin B}{\sin C} = \frac{BP}{PC}\cdot\frac{b}{c}