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anemone
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MHB
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Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area.
Awesome, MarkFL, awesome!(Sun) This is your second time answered to my challenge problem and thank you for participating!MarkFL said:Here is my solution:
The first thing I notice is that there is cyclic symmetry between $x$ and $y$, and so setting $y=x$, we find:
\(\displaystyle 2x^3+3x^2-1=(x+1)^2(2x-1)=0\)
Thus, we know the points:
\(\displaystyle (x,y)=(-1,-1),\,\left(\frac{1}{2},\frac{1}{2} \right)\)
are on the given curve. Next, if we begin with the line:
\(\displaystyle y=1-x\)
and cube both sides, we obtain:
\(\displaystyle y^3=1-3x+3x^2-x^3\)
We may arrange this as:
\(\displaystyle x^3+3x(1-x)+y^3=1\)
Since $y=1-x$, we may now write
\(\displaystyle x^3+3xy+y^3=1\)
And since the point \(\displaystyle \left(\frac{1}{2},\frac{1}{2} \right)\) is on the line $y=1-x$, we know the locus of the given curve is the line $y=1-x$ and the point $(-1,-1)$. Hence, there can only be one set of points on the given curve that are the vertices of any triangle, equilateral or otherwise.
Using the formula for the distance between a point and a line, we find the altitude of the equilateral triangle will be:
\(\displaystyle h=\frac{|(-1)(-1)+1-(-1)|}{\sqrt{(-1)^2+1}}=\frac{3}{\sqrt{2}}\)
Using the Pythagorean theorem, we find that the side lengths of the triangle must be:
\(\displaystyle s=\frac{2}{\sqrt{3}}h=\sqrt{6}\)
And so the area of the triangle is:
\(\displaystyle A=\frac{1}{2}sh=\frac{1}{2}\sqrt{6}\frac{3}{\sqrt{2}}=\frac{3\sqrt{3}}{2}\)
Pranav said:Why both of you posted the same solution?
MarkFL said:I fixed it. (Hug)
An equilateral triangle is a type of triangle where all three sides are equal in length and all three angles are equal to 60 degrees.
The formula for finding the area of an equilateral triangle is A = (√3/4) x s^2, where A is the area and s is the length of one side.
The formula is derived from the fact that an equilateral triangle can be divided into two congruent right triangles with one side being half the length of the base and the other being the height. The formula for the area of a right triangle is 1/2 x base x height, and when we substitute in the values for an equilateral triangle, we get A = (√3/4) x s^2.
Yes, since all sides and angles of an equilateral triangle are equal, the area will also be the same for all equilateral triangles with the same side length.
The side length of an equilateral triangle can be measured by using a ruler or measuring tape. Alternatively, if you know the perimeter of the triangle, you can divide it by 3 to get the length of one side.