Three points defining a circle | Special properties and parts of triangles | Geometry | Khan Academy

TL;DR

The circumcenter of a triangle can be found by drawing perpendicular bisectors of its sides, and any circle with its vertices on the circumference will have the circumcenter as its center.

Transcript

We know that three points define a triangle. So if I were to take three random points here, so let's call that point A, point B, and then let's say this is point C right over here. If we say that these three points are the vertices of a triangle, they define a unique triangle. So this would be triangle A-- try to draw my lines as straight as possib... Read More

Key Insights

• 🔺 Three non-collinear points can form a unique triangle.
• 🔺 The circumcenter of a triangle is equidistant from its vertices.
• 🙃 Perpendicular bisectors of a triangle's sides intersect at the circumcenter.
• ⭕ Any circle with its vertices on the circumference has the circumcenter as its center.
• ⭕ The circumcenter can be located outside the triangle and still be the center of the circle.

Questions & Answers

Q: How can the circumcenter of a triangle be determined?

The circumcenter of a triangle can be found by drawing perpendicular bisectors of its sides and locating their intersection point, which is the circumcenter.

Q: Do all triangles have a circumcenter?

Yes, every non-collinear triangle has a unique circumcenter.

Q: What is a circumradius?

The circumradius is the distance between the circumcenter of a triangle and any of its vertices. It is equal for all three vertices.

Q: Can a circle with its vertices on the circumference have a circumcenter outside the triangle?

Yes, it is possible for the circumcenter to be located outside the triangle, but it will still be the center of the circle due to its equidistance from the triangle's vertices.

Summary & Key Takeaways

• Three non-collinear points can define a unique triangle, which has a unique circumcenter equidistant from its vertices.

• The circumcenter can be found by drawing perpendicular bisectors of the triangle's sides, which intersect at the circumcenter.

• A circle can be formed with the circumcenter as its center, and the triangle's vertices lying on the circumference.