I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base."
(Figure is in the link) http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII20.html
English isn't my first language, so I just want to make sure that I understood something correctly. We prove the theorem by putting the two angles one on the other for the circumference. I was just wondering, can I assume that the angles do not need to be one on the other and they can have different portion of the circumference, as long as the circumference are of the same lenght ? (Will the proposition still work in this way?) I guess that Euclid did the proof by putting the angles one on the other for making the demonstration less wordy. (Less long to read)
Thank you!
Yes, you could say the same as long as the circumferences are the same length (arc lengths are the same) .
You could prove this by using both the theorem you link to above and the theorem that congruent arcs have congruent central angles (or congruent central angles have congruent arcs).