A point P moves inside a triangle formed by A(0,0) , B(1,√3.) , C(2,0) satisfying PA<=1 ,PB<=1 and >PC<=1 .If the area bounded by the curve traced by P is equal to aπ/b , then find the minimum value of (a+b).

I have tried to consider P as (h,k) and then related the distances of AP , BP , CP According to the condition given in the question. Since only one option amongst AP<= 1,BP<= 1,CP<= 1 has to be satisfied , therefore I tried considering only one option at a time.

That is:

1) if AP<=1:


But then again , I am overwhelmed by the number of variables in the question since I need to find the locus of the point P and find the area of the curve traced by P. Can Anyone give me a hint on how to solve this problem?

  • $\begingroup$ Have you noticed that triangle ABC is equilateral ? But you should say at once that the set of points M fullfilling PA<1 OR PB<1 OR PC<1. $\endgroup$
    – Jean Marie
    May 17, 2020 at 7:37

1 Answer 1


Traingle $ABC$ is equilateral, which means that all its angles have value $\pi/3$.

The curve traced by $P$ is made of three arcs of circle, one at each vertex delimitating the sixth of a disk with radius 1 (whose area is $\pi$.

Therefore the 3 areas totalize a value $3 \times \pi/6=\pi/2=a\pi/b$ with $a=1$ and $b=2$,

Consequence $a+b=3$, because all other solutions will be written $k \pi/(2k)$ providing a larger value $k+2k=3k$.


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