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Toolkit 108

Ratio Lemma

sin⁡A1sin⁡A2=BPPC⋅bc=BPPC⋅sin⁡Bsin⁡C\frac{\sin A_1}{\sin A_2}=\frac{BP}{PC}\cdot\frac{b}{c}=\frac{BP}{PC}\cdot\frac{\sin B}{\sin C}
Ratio Lemma configuration in triangle ABC

Proof

By Toolkit 107 — Law of Sines in △ABP and △APC,

sin⁡A1BP=sin⁡BAP=sin⁡P1c\frac{\sin A_1}{BP}=\frac{\sin B}{AP}=\frac{\sin P_1}{c}
sin⁡A2PC=sin⁡CAP=sin⁡P2b\frac{\sin A_2}{PC}=\frac{\sin C}{AP}=\frac{\sin P_2}{b}

By dividing,

sin⁡A1BPsin⁡A2PC=sin⁡BAPsin⁡CAP=sin⁡P1csin⁡P2b\frac{\frac{\sin A_1}{BP}}{\frac{\sin A_2}{PC}}=\frac{\frac{\sin B}{AP}}{\frac{\sin C}{AP}}=\frac{\frac{\sin P_1}{c}}{\frac{\sin P_2}{b}}
sin⁡A1sin⁡A2⋅PCBP=sin⁡Bsin⁡C=bc\frac{\sin A_1}{\sin A_2}\cdot\frac{PC}{BP}=\frac{\sin B}{\sin C}=\frac{b}{c}
sin⁡A1sin⁡A2=BPPC⋅sin⁡Bsin⁡C=BPPC⋅bc\frac{\sin A_1}{\sin A_2}=\frac{BP}{PC}\cdot\frac{\sin B}{\sin C}=\frac{BP}{PC}\cdot\frac{b}{c}
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