Two circles for every triangle

The idea

Every triangle has exactly one circle through its three vertices, the circumcircle, and exactly one circle that fits tightly inside it touching its three sides, the incircle.

Around and inside

Take the three corners of a triangle. They are three points that do not lie on one straight line. Two special circles belong to every triangle: one goes around it and one sits inside it.

How many circles pass through all three vertices of a triangle?

• None
• Exactly one
• Many

Draw the circles

Turn on both circles for each triangle. The orange circle goes through the vertices. The green circle fits inside and touches the sides.

One of each

Every triangle has exactly one circle through its three vertices, the circumcircle, and exactly one circle that fits tightly inside it, touching its three sides: the incircle.

The circumcircle goes around the triangle, through A, B and C. The incircle sits inside the triangle and touches each side once. We will use the radius R of the circumcircle and the radius r of the incircle in two new formulas for the area.

Notes

Every triangle has exactly one circumcircle (through its three vertices) and exactly one incircle (inside it, touching its three sides).

Check yourself

Which circle is the circumcircle of a triangle?

How many circles pass through all three vertices of a given triangle?

Which circle is the larger one: the circumcircle or the incircle of the same triangle?

The incircle touches each side of a triangle once. How many sides does it touch in all?

Answer: 3

The incircle touches all 3 sides.

• The circle that passes through all three vertices — correct. Yes. It goes around the triangle.
• The circle inside the triangle that touches its three sides. That is the incircle.
• Any circle that is big enough to hold the triangle. The circumcircle must pass through all three vertices.
• Two. There is only one. The circumcircle is the circle through the three vertices.
• Exactly one — correct. Yes. Given any three points not on a straight line, there is exactly one circle through them.
• As many as we like. Three points that are not on a line fix the circle: there is exactly one.
• The incircle. The incircle sits inside the triangle, which sits inside the circumcircle.
• They are always equal. They are different circles: one goes around the triangle and one fits inside it.
• The circumcircle — correct. Yes. The incircle fits inside the triangle, and the triangle fits inside the circumcircle.