Constructing Points of Concurrency

1
Constructing Points of Concurrency

• Segments, rays, lines, or planes are concurrent
  if they intersect in a single point.
• The angle bisectors of a triangle, perpendicular
  bisectors of each side of a triangle, altitudes
  of a triangle, and medians of a triangle are
  all concurrent with each other.
• The Geometers Sketchpad lab illustrates the four
  points of concurrency associated with triangles.

2
Constructing Points of Concurrency

• C-9 Angle Bisector Concurrency Conjecture
• The three angle bisectors of a triangle are
  concurrent.
• The incenter is the point at which the angle
  bisectors are concurrent.
• C-13 Incenter Conjecture
• The incenter of a triangle is equidistant from
  each side of the triangle.
• The incenter is the center of a circle inscribed
  in the triangle.

3
Constructing Points of Concurrency

• C-10 Perpendicular Bisector Concurrency
  Conjecture
• The three perpendicular bisectors of a triangle
  are concurrent.
• The circumcenter is the point at which the
  perpendicular bisectors are concurrent.
• C-13 Circumcenter Conjecture
• The circumcenter of a triangle is equidistant
  from each vertex of the triangle.
• The circumcenter is the center of a circle
  circumscribed about the triangle.

4
Constructing Points of Concurrency

• C-11 Altitude Concurrency Conjecture
• The three altitudes (or the lines containing the
  altitudes) of a triangle are concurrent.
• The orthocenter is the point at which the
  altitudes are concurrent.