Inscribed Angle Theorem
- Author:
- Brad Ballinger
- Topic:
- Angles, Circle, Geometry, Isosceles Triangles, Triangles
Here's a triangle inscribed in a circle.
What makes it an "inscribed triangle"?
- It's a triangle.
- Each vertex is on the circle.
Use the Angle tool to measure and :
- Click the Angle tool, second from the right, to activate it. Its frame will change color to show that it's active.
- Click A, then B, then C. Each point will grow a "halo" to confirm your click. If you click but don't see the halo, try again.
- Click A, then D, then C. (Notice that the Angle tool stays active - so when you want to do something else, pick another tool.)
Use the Move tool to manipulate the diagram. As you do, compare the measures of and .
- Click the Move tool, leftmost on the toolbar, to activate it. Its frame will change color to show that it's active.
- Click and drag A, B, and/or C. Look for relationships between the angle measures that you see.
What do you notice about the measures of and ?
The Inscribed Angle Theorem is observation #3 in my answer to the question above. Let's prove that it's always true. We'll look at three cases:
Case 1: the center of the circle is on one of the rays (sides) of the inscribed angle.
Case 2: the center of the circle is in the interior of the inscribed angle.
Case 3: the center of the circle is in the exterior of the inscribed angle.
Case 1: D is on a side of the angle.
In the figure below, I've shown the situation where D is on ray BA. The thinking goes the same way if D is on ray BC instead.
Do you see any isosceles triangles? How do you know they are isosceles?
Where do isosceles triangles give us congruent angles?
Same info another way: matching angles are congruent in the figure below.
In the figure above, there are (at least) two ways to add onto the teal angle () to make . Which do you see?
Since , we conclude that . (This is one of the Exterior Angle theorems: for any triangle, one exterior angle equals the sum of the "other two" interior angles.)
Since , we conclude that .
But , so . This proves Case 1.
Case 2: D is between ray BA and ray BC.
Line BE cuts the circle into two halves: one with A; the other with C.
Move A and C freely within their own halves.
You can move B if you like, but make sure that line BE never passes through A or C.
We can apply Case 1 to the tangerine angles. It tells us that . What does Case 1 tell us about the green angles?
How are the purple angles related?
Case 3: D is neither *on* nor *between* rays BA and BC.
Let's reuse the Case 2 figure. Move A, B, and C so that they are still in clockwise order but A and C are on the same side of line BE; that is, line BE does not meet segment .
How are the purple angles related?
I'm putting a link here to a very surprising 5-minute 3B1B video: https://www.youtube.com/watch?v=HEfHFsfGXjs. The sequel to that video uses the Inscribed Angle Theorem to explain what happens in the first video.