Question

Challenge:
Let ABC be an equilateral triangle and P an arbitrary point inside ABC such that :
max{PA,PB,PC} = 1/2(PA+PB+PC)
Find the locus of P.

Answer 1

What does max{PA,PB,PC} implies?

Answer 2

It means the one among PA,PB,PC which has the maximum length.

Answer 3

max{PA,PB,PC} = the maximum distance between P to either of A, B or C
And if P lies on the circumcircle , then one of PA, PB or PC is sum of the other 2.

Answer 4

max{PA,PB,PC} = 1/2(PA+PB+PC)
And this whole equation?

Answer 5

this means that , the max distance from P to any pt ( A, B or C) is equal to half of the sum of all three distances.

Answer 6

im thinking there are no points that fulfill that condition

Answer 7

The locus is the circumcircle of ABC .Can you prove it?

Answer 8

max{PA,PB,PC} = a which makes it lie on the circumcircle. Now as thinker said if one of them lie on circumcircle then the sum of other two is equal to the third.

max{PA,PB,PC} = 1/2 (PA+PB+PC)
a = 1/2 (a+a)

Answer 9

This problem is probably taken from here: http://www.scribd.com/dorh7343/d/23254911-Mathematical-Olympiad-Challenges

Answer 10

I didn't prove it, I know I just did an intuitive proof

Answer 11

ok if it lies on circumcircle then
sqrt(3)x/2 = x

??