In the given diagram, identify the points which are equidistant from point B.

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Solution

If points X and Y are equidistant from point Z then d(X, Z) = d(Y, Z).
In the given question point B is 2 so d(A, B) = 2 - 0 = 2 and d(C, B) = 4 - 2 = 2
$∴$ d(A, B) = d(B, C)

Hence, points A and C are equidistant from point B

Point B is 2 so d(P, B) = 2 - (-2) = 4 and d(D, B) = 6 - 2 = 4
$∴$ d(P, B) = d(D, B)

Hence, points P and D are also equidistant from point B.

Therefore the set of points that are equidistant from point B are:
i) Points A and C
ii) Points P and D

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Mention the proper order of identifying a point D on BC such that $\frac{A r (△ A B D)}{A r (△ A D C)} = \frac{2}{3}$ ?
1) Join $B_{5} C$ and draw a line parallel to $B_{5} C$ from $B_{2}$ .
2) Identify the ratio in which the point D divides BC using the relation given.
3) Mark 5 points B₁, B₂, B₃, B₄ and B₅ on BX such that they are equidistant.
4) Construct a ray BX which makes an acute angle with line segment BC.
5) The point of intersection of the parallel line from B₂ with BC is the point D.

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