Equidistance—A Focus on Distance
Lesson 12Topics
x Equidistance between points.
x Equidistance between lines.
x Circumcircles of triangles.
x Incircles of triangles.
x The Euler line.
'H¿QLWLRQV
x circumcenter of a triangle: The center of the circumcircle of a triangle is its circumcenter. It is the
location where the three perpendicular bisectors of the triangle coincide.
x circumcircle of a triangle: For each triangle, there is a unique circle that passes through the vertices of
the triangle. This circle is the circumcircle of the triangle.
x equidistant: A point is said to be equidistant from two or more objects if its distance from each of
those objects is the same.
x incenter of a triangle: The center of the incircle of a triangle is its incenter. It is the location where the
three angle bisectors of the triangle coincide.
x incircle of a triangle: For each triangle, there is a unique circle sitting inside the triangle tangent to
each of its three sides. This circle is the incircle of the triangle.
Results
x The set of points equidistant from a pair of points A and B is precisely the set of points on the
perpendicular bisector of AB.
x The three perpendicular bisectors of a triangle coincide at a single point. This point is equidistant from
all three vertices of the triangle and, therefore, lies at the center of a circle that passes through all three
vertices of the triangle.