A Geometry Problem Revised
by luimichael, Nov 9, 2007, 1:17 PM
Given that ABC is an equilateral triangle of unknown side. P is a point inside triangle ABC with PA = 4, PB = 3 and PC = 5.
Find the size of the equilateral triangle.
Solution:
Following the idea in the above solution, suppose the problem is defined in a more general setting as follows:
Given four points P, A,B and C in the same plane with PA = a, PB = b, PC = c and AB =BC =CA = x.
Determine the value of x in terms of a, b and c.
The solution is neat:
,
where
represents the area of triangle with sides a, b and c.
The plus or minus sign is determined by the following rule: Take a plus sign if the point P is inside triangle ABC and of course take a minus sign if P is outside triangle ABC.
Proof:
Alternative proof:
Further questions:
1. What happens if P is outside triangle ABC ?
2. What is the formula for the general case?
That is to say, what is the expression for x if the lengths PA, PB and PC are given to be a, b and c respectively?
3. How comes the solution of this problem is apparently the same as that of the Fermat Point Problem
Answer:
Find the size of the equilateral triangle.
Solution:
{Note that triangle PAQ is a right-angled triangle}
.Following the idea in the above solution, suppose the problem is defined in a more general setting as follows:
Given four points P, A,B and C in the same plane with PA = a, PB = b, PC = c and AB =BC =CA = x.
Determine the value of x in terms of a, b and c.
The solution is neat:
,where
represents the area of triangle with sides a, b and c.The plus or minus sign is determined by the following rule: Take a plus sign if the point P is inside triangle ABC and of course take a minus sign if P is outside triangle ABC.
Proof:
.
.
![$ + [2(a^2b^2 + b^2c^2 + c^2a^2) - (a^4 + b^4 + c^4)]$](http://latex.artofproblemsolving.com/3/c/d/3cdf41dd19b9c7bacc9721d25625f009bf7665ab.png)
Alternative proof:
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![$ (t^2 - \frac {a^2 + b^2 + c^2}2 )^2 = \frac 3 4 [2(a^2b^2 + b^2c^2 + c^2a^2 ) - (a^4 + b^4 + c^4)]$](http://latex.artofproblemsolving.com/2/3/1/231b5f0e13d84e22d8c165b260ff06ee13598d82.png)
,
,
,Further questions:
1. What happens if P is outside triangle ABC ?
2. What is the formula for the general case?
That is to say, what is the expression for x if the lengths PA, PB and PC are given to be a, b and c respectively?
3. How comes the solution of this problem is apparently the same as that of the Fermat Point Problem
Answer:
September 2020
January 2020