What types of triangles are constructible?

Question

What types of triangles are constructible?

I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other triangles be constructed?

Answer 1

A triangle is constructible if an only if its sides are constructible. Constructible lengths is a solved problem so your question about triangles is as well.

Given a line segment, we can construct other segments if the ratio between the two segments can be gotten from adding, subtracting, multiplying, and dividing two numbers and taking the square root of one number, starting from the number $1$.

So by adding $1$ to itself repeatedly we get all natural numbers ratios. By dividing we get all rational ratios. By taking square roots we get many more ratios.

Many ratios are still not possible, such as those gotten by trisecting the $60°$ angle in an equilateral triangle. Both those triangles that are constructible give us much variety for creativity. I particularly like the Golden triangle, which is contained in the pentagon and pentagram.

Answer 2

There are of course infinitely many constructible triangles. However, most triangles are actually not constructible. A triangle is constructible iff all of its side lengths are real constructible numbers (and they satisfy the triangle inequality, of course). The constructible numbers are all the numbers you can get from $\mathbb{Q}$ by a finite number of additions, subtractions, multiplications, divisions, square roots and complex conjugations, so they include things like $\sqrt{2}$, but not transcendal numbers like $\pi$. Not even all algebraic numbers are constructible; for example $e^{\frac{2\pi}{7}i}$ is not constructible, even though it satisfies $z^7-1=0$, which is why you cannot construct a regular heptagon, or a triangle with any angle of $\frac{2\pi}{7}$ radians.