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Question: If I is the center of a circle inscribed in a triangle ABC, then \[\left| \overrightarrow{BC} \ri...

If I is the center of a circle inscribed in a triangle ABC, then
BCIA+CAIB+ABIC\left| \overrightarrow{BC} \right|\overrightarrow{IA}+\left| \overrightarrow{CA} \right|\overrightarrow{IB}+\left| \overrightarrow{AB} \right|\overrightarrow{IC} is
(A) 0\overline{0}
(B) IA+IB+IC\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}
(C) IA+IB+IC3\dfrac{\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}}{3}
(D) IA+IB+IC2\dfrac{\overrightarrow{IA}+\overrightarrow{IB}+\overrightarrow{IC}}{2}

Explanation

Solution

Get the position of the vertices A, B, and C of ΔABC\Delta ABC with respect to the incenter I. Use the formula for the position vector of incenter of ΔABC\Delta ABC , BC.a+CA.b+AB.cBC+CA+AB\dfrac{BC.\overrightarrow{a}+CA.\overrightarrow{b}+AB.\overrightarrow{c}}{BC+CA+AB} where a\overrightarrow{a} , b\overrightarrow{b} , and c\overrightarrow{c} are the affixes of vertices of ΔABC\Delta ABC and get the position vector of I. Since the calculated position vector is with respect to I so, the position vector of I with respect to I must be equal to zero. Now, solve it further and calculate the value of BCIA+CAIB+ABIC\left| \overrightarrow{BC} \right|\overrightarrow{IA}+\left| \overrightarrow{CA} \right|\overrightarrow{IB}+\left| \overrightarrow{AB} \right|\overrightarrow{IC} .

Complete step by step solution:
According to the question, we are given that I is the center of the circle inscribed in the triangle ABC and we are asked to find the possible value of the expression, BCIA+CAIB+ABIC\left| \overrightarrow{BC} \right|\overrightarrow{IA}+\left| \overrightarrow{CA} \right|\overrightarrow{IB}+\left| \overrightarrow{AB} \right|\overrightarrow{IC} .
First of all, let us assume that I is the incenter of ΔABC\Delta ABC .

From the above diagram, we can observe that
The position vector of vertex A with respect to the incenter I = IA\overrightarrow{IA} …………………………………………….(1)
The position vector of vertex B with respect to the incenter I = IB\overrightarrow{IB} ……………………………………….……(2)
The position vector of vertex C with respect to the incenter I = IC\overrightarrow{IC} …………………………………………….(3)
We know the formula for the incenter of ΔABC\Delta ABC , BC.a+CA.b+AB.cBC+CA+AB\dfrac{BC.\overrightarrow{a}+CA.\overrightarrow{b}+AB.\overrightarrow{c}}{BC+CA+AB} where a\overrightarrow{a} , b\overrightarrow{b} , and c\overrightarrow{c} are the affixes of the vertices of ΔABC\Delta ABC ………………………………………..(4)
Now, from equation (1), equation (2), equation (3), and equation (4), we get
The affix of the incenter of ΔABC\Delta ABC = BCIA+CAIB+ABICBC+CA+AB\dfrac{BC\left| \overrightarrow{IA} \right|+CA\left| \overrightarrow{IB} \right|+AB\left| \overrightarrow{IC} \right|}{BC+CA+AB} ………………………………..(5)
But the position vector of I with respect to I must be equal to zero ……………………………………..(6)
Now, from equation (5) and equation (6), we get
0=BC.IA+CA.IB+AB.ICBC+CA+AB\Rightarrow \overrightarrow{0}=\dfrac{BC.\overrightarrow{IA}+CA.\overrightarrow{IB}+AB.\overrightarrow{IC}}{BC+CA+AB}
0=BC.IA+CA.IB+AB.IC\Rightarrow \overrightarrow{0}=BC.\overrightarrow{IA}+CA.\overrightarrow{IB}+AB.\overrightarrow{IC} ……………………………………………….(7)
BC, CA, and AB can also be written as BC\left| \overrightarrow{BC} \right| , CA\left| \overrightarrow{CA} \right| , and AB\left| \overrightarrow{AB} \right| …………………………………………..(8)
Now, from equation (7) and equation (8), we get
0=BCIA+CAIB+ABIC\Rightarrow \overrightarrow{0}=\left| \overrightarrow{BC} \right|\overrightarrow{IA}+\left| \overrightarrow{CA} \right|\overrightarrow{IB}+\left| \overrightarrow{AB} \right|\overrightarrow{IC}
So, the correct answer is “Option A”.

Note: For this type of question, one must remember the formula for the position vector of the incenter of any triangle. That is the position vector of incenter of ΔABC\Delta ABC , BC.a+CA.b+AB.cBC+CA+AB\dfrac{BC.\overrightarrow{a}+CA.\overrightarrow{b}+AB.\overrightarrow{c}}{BC+CA+AB} where a\overrightarrow{a} , b\overrightarrow{b} , and c\overrightarrow{c} are the affixes of vertices of ΔABC\Delta ABC .