Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:
"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90o- 1/2*arc ED.
Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90o- 1/2*arc ED.
=> Angle A = Angle EFC.
Angle A + Angle EFB = Angle EFC + Angle EFB = 180o.
Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Tom

2. ## Re: Chordal quadrileteral problem

Originally Posted by tomkoolen
Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:
"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90o- 1/2*arc ED.
Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90o- 1/2*arc ED.
=> Angle A = Angle EFC.
Angle A + Angle EFB = Angle EFC + Angle EFB = 180o.
Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Tom

3. ## Re: Chordal quadrileteral problem

It is in America more often called cyclic quadrilateral; a quadrilateral with all of its vertices on a circle.

4. ## Re: Chordal quadrileteral problem

Hi,

In a circle an arc angle is the length of the arc divided by the radius of the circle (formula $s=r\theta$). Equivalently, an arc angle is the measure of the "central" angle; i.e. the angle from the center of the circle with endpoints the endpoints of the arc. Here's the theorem which is the first line of your proof: