The Other Proofs You've Seen Are Wrong | Complete Circle Theorem Proofs

If you’ve studied maths at school, you’ll have come across the circle theorems, like the angle at the centre is double the angle at the circumference, and angles in a cyclic quadrilateral sum to 180 degrees and the alternate segment theorem.

If you had a good teacher they might even have seen a proof of why these theorems are true. But most of the time the proofs given for these theorems are actually incomplete, and in this video I’ll show you where they usually fall short and how to fix them, giving full proofs of 5 of the most important circle theorems.

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[Fun_maths]

There is a notion of directional angles which are defined as the angle you need to rotate line l1 counter clockwise (of clockwise if you like) to get line l2, notive then that the angle is no longer symmetric, instead we have
angle(l1, l2)=-angle(l2, l1)
angle(l1, l2)+angle(l2, l3)+angle(l3, l1)=0
(you can add 180 if it makes you feel better but rotating a line 180 degrees does not change its direction)
and then you don't have to check for all cases, just make sure that you have all the directions right, so for the first theorem it will look like (I will use a as short for angle):
a(AB, AC)=a(AB, AO)+a(AO, AC)=x+y
a(BO, CO)=a(BO, AO)+a(AO, CO)=a(BO, AB)+a(AB, BO)+a(AO, AC)+a(AC, CO)=x+x+y+y=2a(AB, AC)
And this works regardless of where a is in the circle, it will just mean that some of the angles might be nagtive but the underlying computation is unchanged.

[AltF-ddzm]

Wish I discovered your channel earlier, actual W of a video