Fix the mistake: Properties of Geometric Figures ① (Triangles)

Five Centers of a Triangle (Centroid, Circumcenter, Incenter, Orthocenter)

A triangle has five important points. Learn the definition and properties of each precisely.

Centroid G

The intersection of the three medians (segments connecting vertices to the midpoints of the opposite sides).
AG:GM = 2:1 (The distance from a vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.)

Example ① (Coordinates of the Centroid)
Find the coordinates of the centroid G of A(0,0), B(6,0), and C(0,6).
Solution:G = ((0+6+0)/3, (0+0+6)/3) = (2, 2)

Circumcenter O

The intersection of the perpendicular bisectors of the three sides → the center of the circumcircle (OA=OB=OC=the circumradius R)
Acute triangle: inside, right triangle: midpoint of the hypotenuse, obtuse triangle: outside

Incenter I

The intersection of the bisectors of the three interior angles → the center of the incircle (inradius r)
r = S/s (S: area of the triangle, s: semiperimeter=(a+b+c)/2)

Example ② (Inradius)
Find the inradius r of a right triangle with side lengths 3, 4, and 5.
Area S = (1/2)×3×4 = 6, semiperimeter s = (3+4+5)/2 = 6
Solution:r = S/s = 6/6 = 1

Orthocenter H

The intersection of the perpendiculars (altitudes) drawn from the three vertices to the opposite sides.
Acute triangle: inside, right triangle: the right-angle vertex, obtuse triangle: outside

Key Points
  • Centroid: intersection of the medians, AG:GM=2:1
  • Circumcenter: intersection of the perpendicular bisectors, center of the circumcircle
  • Incenter: intersection of the angle bisectors, r=S/s
  • Orthocenter: intersection of the altitudes
  • In an equilateral triangle, all four points coincide

Practice Problems

  1. Find the coordinates of the centroid G of A(2,0), B(8,0), and C(4,6).
  2. Find the inradius r of a right triangle with side lengths 5, 12, and 13.
  3. In an acute triangle, state which of the circumcenter and orthocenter lies inside the triangle.

Solutions and Explanations

  1. G = ((2+8+4)/3, (0+0+6)/3) = (14/3, 2)
  2. S=(1/2)×5×12=30、s=(5+12+13)/2=15 → r=30/15=2
  3. In an acute triangle, both the circumcenter and orthocenter lie inside the triangle.

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