Does such an Isosceles triangle exist?

Question

What isosceles triangle meets the following requirement:

• the unique side has a length of 1
• the unique angle can be multiplied by a whole number to equal 180°
• the similar side lengths are whole numbers
• the triangle is not equilateral

If it is possible and there are multiple, what is the triangle with the greatest unique angle?

Answer 1

You wrote two conditions for the unique side length. I assume you meant that length to be $1$, and the equal sides to be of integer length.

Let the unique angle be $2\varphi$ so you can easily split the triangle into two congruent right-angled triangles. Then the ratio between half the unique side length and the equal side length $a$ is

$$\sin\varphi=\frac{\,\frac12\,}a\quad\Rightarrow\quad\csc\varphi=2a$$

Now consider Niven's theorem. It states that

$$\sin\varphi=x\text{ with } \varphi\in\pi\mathbb Q,x\in\mathbb Q,0\le\varphi\le90° \\\Rightarrow\quad (\varphi,x)\in\{(0°,0),(30°,\tfrac12),(90°,1)\}$$

So simply from the fact that your solution would have to satisfy both $\varphi\in\pi\mathbb Q$ and $\sin\varphi=\tfrac1{2a}\in\mathbb Q$ along with $0<\varphi<90°$ you can see that there can be no solution satisfying the requirements.