Definition
Lines are concurrent if they intersect at exactly one point. In a triangle, four specific sets of segments (bisectors, medians, altitudes) are guaranteed to be concurrent.
Why It Matters
Every triangle has “secret” points where different priorities are perfectly balanced. If you need to find the point of maximum mass balance (Centroid) or maximum clearance from the sides (Incenter), you must use this math. These aren’t just academic curiosities; they are the “Geometric Solutions” to optimization problems in logistics, civil planning, and structural engineering.
Core Concepts
- Incenter:
- Intersection of the three Angle Bisectors.
- Property: Equidistant from the sides.
- Function: Center of the incircle (inscribed circle).
- Circumcenter:
- Intersection of the three Perpendicular Bisectors of the sides.
- Property: Equidistant from the vertices.
- Function: Center of the circumcircle (circumscribed circle).
- Centroid:
- Intersection of the three Medians.
- Property: The “Center of Gravity.” Located 2/3 the distance from vertex to opposite midpoint.
- Orthocenter:
- Intersection of the three Altitudes.
- Note: In an obtuse triangle, the orthocenter lies outside the triangle.