Multiple choice

In $\Delta ABC, x,y$ and $z$ are the distances of incentre from angular points $A,B,$ and $C$ respectively. If $\displaystyle \frac{xyz}{abc}=\displaystyle \frac{\lambda r}{s}$, then $\lambda=$

  1. $1$
  2. $2$
  3. $3$

  4. none of these
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A Correct answer
Explanation

In a triangle, the distance from the incentre to the vertices are x = r/sin(A/2), y = r/sin(B/2), z = r/sin(C/2). The product xyz = r^3 / (sin(A/2)sin(B/2)sin(C/2)). Using the identity sin(A/2)sin(B/2)sin(C/2) = r/(4R), we get xyz = 4Rr^2. Also, abc = 4Rrs. Thus, xyz/abc = (4Rr^2) / (4Rrs) = r/s. Therefore, lambda = 1.

AI explanation

Using the standard formula for the distance to the incenter from a vertex, we have x = r times the cosecant of A divided by 2, y = r times the cosecant of B divided by 2, and z = r times the cosecant of C divided by 2. The product xyz equals r^3 divided by the product of the sine halves. Using the identity for sine halves in a triangle, xyz divided by abc equals r divided by 4R^2 times s. This simplifies to r divided by s, showing that lambda equals 1.

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