Given △ABC, BC=7, AC=5, AB=8, P is any point in theplane, find
1. the minimum of PA²+PB²+PC² and the corresponded value of PA.
2. the minimum of PA+PB+PC and the correspondedvalue of PA.
A simple math.
2012-05-19 6:44 am
回答 (4)
2012-05-22 5:13 am
2012-05-20 8:11 pm
(1) Here is a general fact.
Let G be the centroid. Then for any P,
(PA)^2 + (PB)^2 + (PC)^2 - 3(PG)^2 = {(GA)^2 + (GB)^2 + (GC)^2} = {(a)^2 + (b)^2 + (c)^2}/3, which is a constant.
Choose P = G, and we get the minimum of (PA)^2 + (PB)^2 + (PC)^2 to be {(a)^2 + (b)^2 + (c)^2}/3.
2012-05-20 12:13:06 補充：
(2) 是費馬點。有很多找尋此點的方式。
Let G be the centroid. Then for any P,
(PA)^2 + (PB)^2 + (PC)^2 - 3(PG)^2 = {(GA)^2 + (GB)^2 + (GC)^2} = {(a)^2 + (b)^2 + (c)^2}/3, which is a constant.
Choose P = G, and we get the minimum of (PA)^2 + (PB)^2 + (PC)^2 to be {(a)^2 + (b)^2 + (c)^2}/3.
2012-05-20 12:13:06 補充：
(2) 是費馬點。有很多找尋此點的方式。
2012-05-20 1:30 am
No special purpose for A=60.
2012-05-19 9:31 pm
1)P 為重心
2)注意∠A = 60° , 所以 P 為外心
2012-05-19 13:51:00 補充：
2)又好像P不是外心~
2012-05-19 14:00:54 補充：
∠A = 60°用意何在??
2)注意∠A = 60° , 所以 P 為外心
2012-05-19 13:51:00 補充：
2)又好像P不是外心~
2012-05-19 14:00:54 補充：
∠A = 60°用意何在??
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