Two Conditions for a Triangle to Be Equilateral


Equivalence in triangle

(The problem uses notations common in triangle geometry.)

Note that the proofs below in fact show more, viz., that both conditions only hold for equilateral triangles, which makes them automatically equivalent.

Solution 1

Adding up the three equations in $\mathbf{P},\;$




which we rewrite as

$\displaystyle 3n^2+\left(2\sum_{cycl}a\right)n+\sum_{cycl}a^2=0.$

Since, it is given that the equation has a solution in integers, hence in reals, the discriminant $\Delta\;$ of this quadratic (in $n)\;$ equation is not negative

$\displaystyle\Delta =4\left(\sum_{cycl}a\right)^2-12\sum_{cycl}a^2\ge 0.$

This is equivalent to

$\displaystyle\sum_{cycl}a^2+2\sum_{cycl}ab-3\sum_{cycl}a^2\ge 0,$

which reduces to $\displaystyle\sum_{cycl}ab\ge\displaystyle\sum_{cycl}a^2,\;$ but, by say, the Rearrangement inequality, we have $\displaystyle\sum_{cycl}ab\le\displaystyle\sum_{cycl}a^2,\;$ implying that $\displaystyle\sum_{cycl}ab = \displaystyle\sum_{cycl}a^2.\;$ From here $\displaystyle\sum_{cycl}(a-b)^2=0,\;$ so that $a=b=c.$

For $\mathbf{Q},\;$ we have


with equality only when $a=b=c.$

Solution 2

For any $n\in\mathbb{Z}^{*},\;$ the graph of the function $y=f(x)=x^2+(2n+1)x+n^2\;$ is tangent to the diagonal $y=x\;$ for $x=-n,\;$ otherwise being above the diagonal.

Convergent iterations without loops

Iterations $x_{k+1}=f(x_k)\;$ converge to the point of tangency, never forming a $3-\mbox{loop}.\;$ It follows, therefore, from the definition of $\mathbf{P}\;$ that it may only hold when $a=b=c.$

Concerning $\mathbf{Q},\;$ with $2p=a+b+c,\;$ we have, say, by the AM-GM inequality,


by Heron's formula. (The equality is only for $b=c.)\;$ On the other hand, say $\displaystyle r_a=\frac{S}{p-a}.\;$ Thus,


But then, by the Cauchy-Schwarz inequality, we get

$\displaystyle\sum_{cycl}\ell_a\le\sum_{cycl}\sqrt{r_br_c}\le r_a+r_b+r_c,$

where the first inequality becomes equality only for $a=b=c\;$ while the second is the equality only for $r_a=r_b=r_c,\;$ which is the same.


The above problem from the Romanian Mathematical Magazine and a solution (Solution 1 by Soumava Chakraborty, Kolkata, India) has been kindly posted at the CutTheKnotMath facebook page by Dan Sitaru.


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  • Seven Problems in Equilateral Triangle
  • Spiral Similarity Leads to Equilateral Triangle
  • Parallelogram and Four Equilateral Triangles
  • A Pedal Property in Equilateral Triangle
  • Miguel Ochoa's van Schooten Like Theorem
  • Incircle in Equilateral Triangle
  • When Is Triangle Equilateral: Marian Dinca's Criterion
  • Barycenter of Cevian Triangle
  • Excircle in Equilateral Triangle
  • Converse Construction in Pompeiu's Theorem
  • Wonderful Trigonometry In Equilateral Triangle
  • 60o Angle And Importance of Being The Other End of a Diameter
  • One More Property of Equilateral Triangles
  • Van Khea's Quickie
  • Equilateral Triangle from Three Centroids
  • Systems of Iterated Equations

    1. Iterations on Monotone Functions
    2. Graphing Equations Is Useful
    3. Graphing Equations Is Useful, II
    4. Graphing Equations Is Useful, III
    5. Graphing Equations Is Useful, IV
    6. Graphing Equations Is Useful, V
    7. Tangential Chaos
    8. Equation in Radicals as a System of Equations
    9. Two Conditions for a Triangle to Be Equilateral

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