Thales's theorem

[Loki123]

I came across a problem that said a triangle with a right angle is inscribed in the circle. The solution says that because it's a triangle with a right angle in the circle, hypotenuse equals the diameter. Searching about this, since I did not know about it before, I found Thales's theorem which states (according to wiki):

In geometry, Thales' theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle.

Now this got me confused. According to this because AC is a diameter the angle ABC is a right angle. While my problem states that there is a right angle inscribed in the circle, with no mention of the diameter. The definition suggests that the angle is a right angle if AC is a diameter, but not that necessarily, a right angle has a diameter as hypotenuse. I am confused about this. Did I even find the right theorem? And If I did, what is the proof for it?

 

[Dr.Peterson]

I came across a problem that said a triangle with a right angle is inscribed in the circle. The solution says that because it's a triangle with a right angle in the circle, hypotenuse equals the diameter. Searching about this, since I did not know about it before, I found Thales's theorem which states (according to wiki):

In geometry, Thales' theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle.

Now this got me confused. According to this because AC is a diameter the angle ABC is a right angle. While my problem states that there is a right angle inscribed in the circle, with no mention of the diameter. The definition suggests that the angle is a right angle if AC is a diameter, but not that necessarily, a right angle has a diameter as hypotenuse. I am confused about this. Did I even find the right theorem? And If I did, what is the proof for it?

Your problem uses the CONVERSE of Thales' Theorem:

Thales's theorem - Wikipedia

en.wikipedia.org

You just had to look further down the page. (The proofs of both are also on the page. And you should explore some links, such as to the inscribed angle theorem.)

But it's great that you recognized that the theorem itself didn't apply to your problem; many students don't notice the difference between a theorem and its converse!

 

[blamocur]

To me the simplest approach is to remember that the angle of an inscribed triangle is 1/2 of the angle of the opposiing arc. For diameters the arc angle is half of the circle, i.e. [imath]180^\circ[/imath], which means that corresponding angle in the triangle is [imath]90^\circ[/imath].

 

[Steven G]

I do agree with Dr Peterson that it's great that you recognized that the theorem itself didn't apply to your problem; many students don't notice the difference between a theorem and its converse!

Very good!