## O≡(0,0), A≡(2,0)AND B≡(1,√3). P(x,y)

consider a triangle OAB formed by O≡(0,0), A≡(2,0)AND B≡(1,√3). P(x,y) is an arbitrary interior point of the triangle moving in such a way that the sum of its distances from the three sides of the triangle is √3 units . find the area of the region representing possible positions of the point P .

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 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor Hi abhishek! (this is an equilateral triangle, of course, and one of the possible positions of P is its centroid) Show us what you've tried, and where you're stuck, and then we'll know how to help!
 Spoiler the triangle is an equilateral triangle, so the distance from any arbitrary point P inside the triangle to the sides of the triangle is constant (and equal to the height of the triangle, so √3 units in this case). proof: consider P, an arbitrary point inside the triangle. the area of the triangle OAB is √3 units and it is also the sum of the areas of the triangles POA= (d1*0A)/2, POB=(d2*OB)/2 and PAB=(d3*AB)/2. but OA=OB=AB=2 units. So the sum of the areas of the smaller triangles is d1+d2+d3=√3. Hence, the sum of the distances from P to the sides of the triangle is constant and equal to √3. q.e.d. so the area of the region representing possible positions of the point P that satisfy the given positions is the area of the triangle = √3 square units...