Carnot's theorem (inradius, circumradius)
Carnot's theorem is a math rule that helps us figure out the relationship between the radii of different shapes that fit inside or around a triangle.
Now, a triangle is a shape that has three sides and three corners. You might have seen shapes like this before - they look like a slice of pizza!
Let's talk about radii. Radii are imaginary lines that go from the center of a circle to its edge. So if you had a circle, the radius would be the line from the middle of the circle to the edge.
Okay, now back to Carnot's theorem. It tells us that if we have a triangle and we draw circles inside and outside of the triangle, the radii of those circles have a special relationship.
The circle on the inside is called the in-circle and the line from the center of that circle to the triangle's corners is called the inradius. The circle on the outside is called the circumcircle and the line from the center of that circle to the triangle's corners is called the circumradius.
So, what is the special relationship between the radii? Carnot's theorem says that the inradius plus the circumradius equals the distance between the three corners of the triangle.
In simpler terms, if we add up the length of the line from each corner of the triangle to the center of the in-circle and the length of the line from each corner of the triangle to the center of the circumcircle, we get the same number as if we just measured the distance between the three corners of the triangle!
That's Carnot's theorem in a nutshell - it's a way to figure out the relationship between the radii of circles that fit inside and around a triangle.
Now, a triangle is a shape that has three sides and three corners. You might have seen shapes like this before - they look like a slice of pizza!
Let's talk about radii. Radii are imaginary lines that go from the center of a circle to its edge. So if you had a circle, the radius would be the line from the middle of the circle to the edge.
Okay, now back to Carnot's theorem. It tells us that if we have a triangle and we draw circles inside and outside of the triangle, the radii of those circles have a special relationship.
The circle on the inside is called the in-circle and the line from the center of that circle to the triangle's corners is called the inradius. The circle on the outside is called the circumcircle and the line from the center of that circle to the triangle's corners is called the circumradius.
So, what is the special relationship between the radii? Carnot's theorem says that the inradius plus the circumradius equals the distance between the three corners of the triangle.
In simpler terms, if we add up the length of the line from each corner of the triangle to the center of the in-circle and the length of the line from each corner of the triangle to the center of the circumcircle, we get the same number as if we just measured the distance between the three corners of the triangle!
That's Carnot's theorem in a nutshell - it's a way to figure out the relationship between the radii of circles that fit inside and around a triangle.