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Show, by vector methods, that the angula...

Show, by vector methods, that the angularbisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

Text Solution

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The correct Answer is:
`I= (alpha overset(to)(a) + beta overset(to)(b) + gamma overset(to)( c))/( alpha + beta + gamma)`
Let AD be the angular bisector of angle A. Let BC , AC and AB are `alpha,beta` and `gamma` respectively . Then `,(BD)/(DC) = (gamma)/(beta)`
Hence position vector of `D= (gamma vec(c ) + beta vec(b))/(gamma+ beta) ` . On AD , there lies a point I which divides it in ratio `gamma+ beta : alpha`
Now position vector of `I = (alpha vec(a) + beta vec(b) + gamma vec(c ))/(alpha+ beta + gamma)`
Which is symmetric in `vec(a) , vec(b) , vec( c) alpha, beta " and " gamma`

Hence I lies on every angle bisector and angle bisectors are concurrent .
Here `alpha= |vec(b)-vec( c)| , beta = |vec(a) - vec(c ) | , gamma= | vec(a) - vec(b)|.`
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