Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
In $\triangle ABC$, if $(II_1)^2 + (I_2I_2)^2 = \lambda R^2$, where $I$ denotes incentre; $I_1, I_2$ and $I_3$ denote centres of the circles inscribed to the sides $BC, CA$ and $AB$ respectively and $R$ be the radius of the circum circle of $\triangle ABC$. Find $\lambda$.
Step-by-Step Solution
Key Concept: The orthocenter distance and excircle radius relationships combined with pedal triangle properties yield the parameter $\lambda$.
Using the property $\sin\frac{B}{2} = \frac{r}{IB}$ and $IB = 4R\sin\frac{A}{2}\sin\frac{C}{2}$, we derive $BI_1 = 4R\sin\frac{A}{2}\cos\frac{C}{2}$. From the pedal triangle property, $I_2I_3 = 4R\cos\frac{A}{2}$. Using $(H_1)^2 = (BI)^2 + (BI_1)^2 = 16R^2\sin^2\frac{A}{2}$ and $(I_2I_3)^2 = 16R^2\cos^2\frac{A}{2}$, we obtain $\lambda = 16$ by comparing both expressions.
Correct Answer: 16