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anonymous
 4 years ago
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.
anonymous
 4 years ago
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.

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0What does max{PA,PB,PC} implies?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0It means the one among PA,PB,PC which has the maximum length.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0max{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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0max{PA,PB,PC} = 1/2(PA+PB+PC) And this whole equation?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0this 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.

dumbcow
 4 years ago
Best ResponseYou've already chosen the best response.0im thinking there are no points that fulfill that condition

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0The locus is the circumcircle of ABC .Can you prove it?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0max{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)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0This problem is probably taken from here: http://www.scribd.com/dorh7343/d/23254911MathematicalOlympiadChallenges

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I didn't prove it, I know I just did an intuitive proof

dumbcow
 4 years ago
Best ResponseYou've already chosen the best response.0dw:1328360697679:dw ok if it lies on circumcircle then sqrt(3)x/2 = x ??

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Thanks for the link foolformath

dumbcow
 4 years ago
Best ResponseYou've already chosen the best response.0@Aron sorry but could you elaborate, im not familiar with that Theorem i dont see how any point on the circumcircle meets the requirement of one length being the sum of the other 2 lengths
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