The parameters used to uniquely identify a triangle?

Question

Given only the radius, only the area, or only the circumference of a circle, the circle is uniquely defined. However, given only the area a rectangle is not uniquely defined. Similarly, how many parameters uniquely define a triangle. One condition is if given all the sides a triangle it is uniquely defined. How many such conditions exist involving the side lengths, the interior angles, the area and the perimeter of the triangle will uniquely define a triangle.

Also when I say uniquely defined please do not consider the orientation or position of the figure in the 2D space.

Answer 1

These are a few ways to uniquely determine a triangle:

1. Given all three side lengths would uniquely identify the triangle.
   This is to the extent that given three positive reals, you can determine whether they form a triangle or not (triangle inequality). If they do, the triangle will be unique.

2. Given two sides length and the angle between them, you can determine the third side (cosine rule) and hence, the triangle.
   This to the extent that given any two positive reals and any $\theta \in (0, \pi)$, you will get a unique triangle.

3. Given a side and the angles that the side makes with the other two. Using this, you can find the third angle and then determine all the sides (sine rule).
   This is to the extent that given any positive real as side and two (positive) angles that sum to $< \pi$, you will get a unique triangle.

Answer 2

• Given the semiperimeter $\rho$, the radius $r$ of the inscribed and the radius $R$ of the circumscribed circle, if all the roots of the cubic equation

\begin{align} x^3-2\rho\,x^2+(\rho^2+r^2+4\,r\,R)\,x-4\rho\,r\,R&=0 \end{align}
are real, they are the side lengths of the corresponding triangle, otherwise no valid triangle exists with given parameters. Also there are some restrictions on the values of $\rho,r,R$, for example, $R\ge2r$.

Hence, if we have the area $S$ of the triangle,

\begin{align} S&=\rho\,r ,\\ \end{align}

we need two more parameters, for example, $r$ and $R$ or $\rho$ and $R$ to find all side lengths.

• Three exradii $r_a,r_b,r_c$ define a unique triangle

\begin{align} a&=\frac{r_a(r_b+r_c)}{\sqrt{r_1r_b+r_br_c+r_cr_a}} . \end{align}

• Given three medians $m_a,m_b,m_c$, \begin{align} a&=\tfrac23\sqrt{2m_b^2+2m_c^2-m_a^2} , \end{align} if $m_a<m_b+m_c$.

• Given three altitudes $h_a,h_b,h_c$

\begin{align} a&= \frac{2\,h_b\,h_c}{% \sqrt{2\,(h_a^2+h_b^2+h_c^2)-((h_a\,h_b/h_c)^2+(h_b\,h_c/h_a)^2+(h_c\,h_a/h_b)^2)} } ,\\ \text{if }\quad \frac1{m_a}<\frac1{m_b}+\frac1{m_c} . \end{align}

• It was proven that any three bisectors $\beta_a,\beta_b,\beta_c$ define a triangle, but in general it's really difficult to find it.

• Also, given three bisected inradii $r_a,r_b,r_c$, the side lengths of corresponding triangle can be uniquely defined, but not any real triplet defines a valid triple of the side length of triangle

\begin{align} r&= \frac{r_a r_b r_c \left(r_a r_b+r_b r_c+r_c r_a +\sqrt{2 r_a r_b r_c (r_a+r_b+r_c) -(r_a^2 r_b^2+r_b^2 r_c^2+r_c^2 r_a^2)}\right) }{ r_a^2 r_b^2+r_b^2 r_c^2+r_c^2 r_a^2 } ,\\ \rho&= \frac{r r_a r_b r_c}{(r-r_a)(r-r_b)(r-r_c)} ,\\ a&=\rho\cdot\frac r{r_a}\Big(2-\frac r{r_a}\Big) . \end{align}