We assumed that there exist four points that have equal angles as shown on picture below, and we want to prove that these four points lie on the one circle by contradiction.

We know that inscribed angles that subtend the same arc on the circle are equal. So we are drawing an additional line and get one more green angle that equal previous two:

So now we have a triangle with interior and exterior angles that are equal. Hence, those two points are the same ones.

]]>Now, I've drawn another angle that also inscribes the yellow arc. This angle is equal to the first yellow angle!

Isn't that neat? This is also how we are able to prove the Power of a Point Theorem. Do you see that since the yellow angles are the same, we can form similar triangles? It's because ---- look! There are two more angles that inscribe the same arc!

The green angles inscribe the same green arc, and so they are the same as well!

Don't worry if it takes a few times of seeing these ideas before they sink in. It's natural when you are learning something unfamiliar. Remember the saying, "You have to spend 10,000 hours doing anything in order to become an expert." So don't worry if takes time, because this is part of the process of attaining real understanding!

]]>You'll see in the lesson that opposite angles of the cyclic quadrilateral add up to \( 180^{\circ}\). There is also something else called Ptolemy's Theorem, which we don't learn in Module 2, but later on in Module 8 (which isn't available yet). This says that the sum of the products of pairs of sides equals the product of the diagonals. Cyclic quadrilaterals are coooool!

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