I couldn't find any sources of this online, so I would like to ask if what I'm proposing below is correct, or if a similar theorem has been proven before.
We know that the angle subtended by the diameter of a circle is always $90^\circ$ (Thales' Theorem). In the image below, the angle at any point $C$ on the highlighted arc will be $90^\circ$ if $AB$ is the diameter.
Suppose now that $AB$ wasn't the diameter. Is there a theorem that says that there does not exist a point $C$ on the circle such that the angle at $C$ is $90^\circ$?
I will use a diagram to explain my reasoning.
My reasoning is as follows: if you could find me a point in the minor arc highlighted above such that $\angle ACB=90^\circ$, then if you 'push' AB down to the diameter to get DE, the $\angle DCE < \angle ACB$, which is a contradiction of Thales' Theorem!
A similar reasoning can be used to explain for the major arc of the circle.
I'm posting this on MSE as I want to know:
- Is this a valid proof? I know it's not rigorous but is the way I'm going about it correct?
- Has this already been proven? Is there a name for this theorem or is it simply an obvious corollary of Thales' theorem (that I wasn't aware of)?
not the prettiest solution but perhaps this helps:
basically I assumed that we have input of 90 degree angle without assuming we are at the center, and we got that $y=x$ thus it indeed must be the center - proving back Thales' theorem