- The sum of the length of two sides of a triangle is larger than the length of the third side.
- An isosceles triangle has the two angles at the base equal to each other.
- Pythagoras Theorem

\[ A = \sqrt{s(s-a)(s-b)(s-c)}\]

where $s$ is the semiperimeter defined as $s = \frac{1}{2}(a+b+c)$. This theorem allows you to calculate the area of a triangle using solely the lengths of the 3 sides. A similar theorem was discovered by the Chinese independently from the Greeks.

Of more recent vintage is Routh's theorem, published in 1896, that describes the ratio between a triangle and another triangle generated via the intersection of 3 cevians.

I was pleasantly surprised to hear about 2 new theorems about triangles that have been proved only relatively recently.

- Conway's little theorem (1976): A triangle is equilateral if and only if the ratio between the length of any 2 sides is rational and the ratio between any 2 angles is rational.
- Grieser and Maronna 2013: Up to congruence, a triangle is determined uniquely by its area, perimeter and the sum of the reciprocals of its angles.

**References:**

- J. Conway, "A Characterization of the Equilateral Triangles and Some Consequences," The Mathematical Intelligencer, 20 March 2014.
- D. Grieser and S. Maronna, "Hearing the Shape of a Triangle," Notices of AMS, vol. 60, no. 11, pp. 1440-1447, 2013.

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