🎯 Triangle Centers

The four classic triangle centers are fundamental to AMC geometry. Each has unique properties and relationships that frequently appear in contest problems.

🎯 The Four Centers

Centroid (G)

Definition: Intersection of the three medians Properties:

  • Divides each median in ratio 2:1 (closer to vertex)
  • Center of mass of the triangle
  • Coordinates: Average of vertex coordinates

Key Formula: If $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, then $G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$

Incenter (I)

Definition: Intersection of the three angle bisectors Properties:

  • Center of the incircle (inscribed circle)
  • Equidistant from all three sides
  • Angle bisector theorem: $\frac{BD}{DC} = \frac{AB}{AC}$

Key Formula: Distance from incenter to side $a$: $r = \frac{A}{s}$ where $A$ is area and $s$ is semi-perimeter

Circumcenter (O)

Definition: Intersection of the three perpendicular bisectors Properties:

  • Center of the circumcircle (circumscribed circle)
  • Equidistant from all three vertices
  • In right triangles, circumcenter is midpoint of hypotenuse

Key Formula: Circumradius $R = \frac{abc}{4A}$ where $A$ is area

Orthocenter (H)

Definition: Intersection of the three altitudes Properties:

  • In right triangles, orthocenter is the right angle vertex
  • In obtuse triangles, orthocenter is outside the triangle
  • Reflect orthocenter across sides to get points on circumcircle

🔗 The Euler Line

Amazing Fact: In any non-equilateral triangle, the centroid, circumcenter, and orthocenter are collinear!

Euler Line Properties:

  • $G$ is between $O$ and $H$
  • $GH = 2 \cdot GO$ (centroid divides in ratio 2:1)
  • Distance relationships: $OH^2 = 9R^2 - (a^2 + b^2 + c^2)$

🔍 Micro-Examples

Centroid Example

In triangle with vertices $(0,0)$, $(6,0)$, $(3,6)$, the centroid is $G = \left(\frac{0+6+3}{3}, \frac{0+0+6}{3}\right) = (3,2)$.

Incenter Example

In triangle with sides 3, 4, 5, the semi-perimeter is $s = 6$, area is $A = 6$, so inradius is $r = \frac{6}{6} = 1$.

Circumcenter Example

In right triangle with legs 3 and 4, the hypotenuse is 5, so circumradius is $R = \frac{5}{2}$ and circumcenter is at midpoint of hypotenuse.

⚠️ Common Traps

Pitfall: Confusing incenter and circumcenter

  • Fix: Incenter is inside, circumcenter can be outside in obtuse triangles

Pitfall: Forgetting Euler line relationships

  • Fix: Remember $GH = 2 \cdot GO$ and all three are collinear

Pitfall: Wrong centroid formula

  • Fix: Centroid is average of coordinates, not sum

🎯 AMC-Style Worked Example

Problem: In triangle $ABC$, the centroid is at $(4,5)$ and one vertex is at $(6,8)$. If the other two vertices are $(2,3)$ and $(x,y)$, find $x + y$.

Solution: Using the centroid formula: $G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$

We have: $(4,5) = \left(\frac{6+2+x}{3}, \frac{8+3+y}{3}\right)$

This gives us:

  • $4 = \frac{8+x}{3} \Rightarrow 12 = 8+x \Rightarrow x = 4$
  • $5 = \frac{11+y}{3} \Rightarrow 15 = 11+y \Rightarrow y = 4$

Therefore, $x + y = 4 + 4 = 8$.

Answer: $x + y = 8$

🔗 Related Topics

💡 Quick Reference

Center Locations

  • Acute triangle: All centers inside
  • Right triangle: Orthocenter at right angle, circumcenter at midpoint of hypotenuse
  • Obtuse triangle: Orthocenter and circumcenter outside

Distance Relationships

  • Centroid: Divides medians 2:1
  • Incenter: Distance to side = inradius
  • Circumcenter: Distance to vertex = circumradius
  • Orthocenter: No simple distance formula

Special Cases

  • Equilateral: All four centers coincide
  • Isosceles: All centers lie on axis of symmetry
  • Right: Orthocenter = right angle vertex

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