Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let ABC be an isosceles triangle with AB=BC . A trisector of B...

Let ABC be an isosceles triangle with $\mathrm{AB}=\mathrm{BC}$. A trisector of $\angle \mathrm{B}$ meets AC at D . If $\mathrm{AB}, \mathrm{AC}$ and $B D$ are integers and $A B-B D=3$, find $A C$.

Step-by-step solution

$$\begin{aligned} & \frac{I_1}{\sin \left(\frac{\pi}{2}+\theta\right)}=\frac{I_3}{\sin \left(\frac{\pi}{2}-3 \theta\right)} \\ & \Rightarrow \text { put } I_3=I_1-3 \\ & \Rightarrow I_1=\frac{3}{4} \operatorname{cosec}^2 \theta \\ & \text { If } \operatorname{cosec}^2 \theta=36 \\ & \text { i.e. } \operatorname{cosec}^2 \theta=6\end{aligned}$$ $$\begin{aligned} & \text { i.e. } \sin \theta=\frac{1}{6} \\ & I_1=\frac{3}{4} \times 36=27 \\ & I_3=24 \\ & \frac{I_1}{\sin \left(\frac{\pi}{2}-\theta\right)}=\frac{D C}{\sin 4 \theta} & \frac{I_1}{\sin \left(\frac{\pi}{2}+\theta\right)}=\frac{A D}{\sin 2 \theta} \\ & \Rightarrow I_2=\frac{I_1 \sin 4 \theta}{\cos \theta}+\frac{I_1 \sin 2 \theta}{\cos \theta} \\ & \Rightarrow I_2=I_1 \times 2 \times \sin 3 \theta\end{aligned}$$ $$\begin{aligned} \Rightarrow & \frac{3}{4} \operatorname{cosec}^2 \theta \times 2 \times \sin 3 \theta \\ \Rightarrow & \frac{3}{2}\left(\frac{3}{\sin \theta}-4 \sin \theta\right) \\ \mathrm{I}_2 & =\frac{3}{2}\left(3 \times 6-\frac{4}{6}\right) \\ & =\frac{3}{2}\left(18-\frac{2}{3}\right) \\ & =\frac{3}{2}\left(\frac{54-2}{3}\right) \\ & =\frac{3}{2} \times \frac{52}{3} \\ \mathrm{I}_2 & =26\end{aligned}$$