How many pieces of information are needed to determine a triangle?

Question

Typically 2 sides and 1 angle need to be given in order to determine a unique triangle. Alternatively 1 side and 2 angles, or the Cartesian coordinates of three vertices, or the area, base, and height. In any case 3 pieces of information.

Is there any way to determine a unique triangle (up to scaling) with just 2 pieces of information? Perhaps in terms of more arcane parameters like the eccentricity of the triangle's inellipse or parameters related to the triangle's 9 point circle or circumcircle, etc.?

Answer 1

As mentioned in my comment, triangles have a special connection with ellipses, namely, for every triangle there is a unique ellipse such that the ellipse circumscribes the triangle, and the triangle circumscribes another ellipse matching the first except smaller by a factor of $\frac12$ in each dimension.

Each ellipse can circumscribe many triangles which circumscribe the quarter-size ellipse, so it is necessary to specify one point on the edge of the main ellipse to determine which triangle is being described. In fact, if the ellipse is specified by the ratio between its radii with center at the origin and by a point on its edge which also corresponds to one vertex of the intended triangle, then the triangle side lengths and angles are completely specified, and the only missing information is the location and orientation of the triangle in the plane.

The mechanism for this transformation is the mentioned "inellipse" also known as the Steiner inellipse and in particular Marden's theorem.

On a side note, every ellipse corresponds by a simple geometric transform to a sine wave, and so the above process could be made even more arcane by specifying a sine wave by the ratio between the amplitude and wavelength and by a point on the curve.

Answer 2

As Andre hinted at in his comment, specifiying two angles of the triangle ($A,B$) will fix the third $C$, thereby specifying the triangle to within a nonzero integer scaling factor:

$$C = \pi - A - B.$$

Answer 3

A triangle can be identified by the coordinates of the three vertices which can be considered a point in $\mathbb R^6$. Take any bijection $\phi\colon \mathbb R^2 \to \mathbb R^6$ and you are done... If you take a bijection from $\mathbb R \to \mathbb R^6$ you can identify a triangle with a single number.

But, if you want some continuity, you need 3 parameters since the space of all triangles, up to isometries, is 3-dimensional.

Answer 4

Here's a quote from the book Geometry Revisited (Coxeter and Greitzer), p. 125-6. I don't know if it answers the question exactly (although it does imply nothing less than 3 pieces of information will work):

"Since a circle may have any radius, and since its center is determined by two coordinates, the set of all circles in the Euclidean plane (and also in the inversive plane) is a three-parameter family, or threefold infinity."

The next part of the paragraph, while not directly relevant to your question, is interesting nonetheless (and hits the 'arcane' part of your question), so I will include it too just for curiosities' sake:

"By interpreting the threefold infinity of circles in the inversive plane as the planes of a three-dimensional space, we could obtain the famous "non-Euclidean" geometry which was discovered independently (between 1820 and 1830) by Gauss, Bolyai, and Lobachevsky. The angles between two intersecting circles appear as the angles between two planes that intersect in a line; two tangent circles appear as two "parallel" planes; and the inversive distance between two non-intersecting circles appears as the distance between two "ultraparallel" planes which have a common perpendicular line, the distance being measured along this line."