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Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

Given 3 point...

Question

Given 3 points with position vectors $_{1},_{2} a n d_{3}$ which form the vertices of a triangle with side lengths $a = |_{2} -_{1} |, b = |_{3} -_{2} |, c = |_{1} -_{3} |$ . Then the in-centre is given by

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Solution

The correct option is B
We know the in-center or inscribed circle centre is given by the weighted average of coordinates by the opposite sides. Here for $_{1}$ opposite side length will be $|_{2} -_{3} |$ which is equal to b. Similarly for $_{2}$ & $_{3}$ it is c and a.
Therefore Incentre will be the point given by the vector $\frac{b_{1} + c_{2} + a_{3}}{a + b + c}$
$∴$ option b is the correct answer.

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Given 3 points with position vectors ¯p1,¯p2 and ¯p3 which form the vertices of a triangle with side lengths a=|¯p2−¯p1|,b=|¯p3−¯p2|,c=|¯p1−¯p3|. Then the in-centre is given by

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