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Triangle Centers Reference (Centroid, Incenter, Circumcenter, Orthocenter) cheat sheet - grade 8-11

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Math Grade 8-11

Triangle Centers Reference (Centroid, Incenter, Circumcenter, Orthocenter) Cheat Sheet

A printable reference covering centroid, incenter, circumcenter, orthocenter, medians, angle bisectors, perpendicular bisectors, and altitudes for grades 8-11.

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Triangle centers are special points where important lines in a triangle meet. This cheat sheet helps students identify the centroid, incenter, circumcenter, and orthocenter and connect each center to its defining construction. These centers appear often in geometry proofs, constructions, coordinate geometry, and problem solving.

Knowing which lines create each center prevents confusion on diagrams and tests.

The centroid is formed by medians and has a useful coordinate formula. The incenter is formed by angle bisectors and is equidistant from the sides of the triangle. The circumcenter is formed by perpendicular bisectors and is equidistant from the vertices.

The orthocenter is formed by altitudes, and its location depends strongly on whether the triangle is acute, right, or obtuse.

Key Facts

  • The centroid is the intersection of the three medians of a triangle, and its coordinate formula is G(x1+x2+x33,y1+y2+y33)G\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right).
  • The centroid divides each median in a 2:12:1 ratio, with the longer part from the vertex to the centroid.
  • The incenter is the intersection of the three angle bisectors and is equidistant from all three sides of the triangle.
  • The incenter is the center of the incircle, and the area can be found with A=rsA=rs, where rr is the inradius and s=a+b+c2s=\frac{a+b+c}{2}.
  • The circumcenter is the intersection of the three perpendicular bisectors and is equidistant from the three vertices.
  • The circumcenter is the center of the circumcircle, so OA=OB=OC=ROA=OB=OC=R when OO is the circumcenter.
  • The orthocenter is the intersection of the three altitudes, where each altitude is perpendicular to the opposite side.
  • In an acute triangle all four centers are inside the triangle, while in an obtuse triangle the circumcenter and orthocenter are outside the triangle.

Vocabulary

Centroid
The point where the three medians of a triangle intersect.
Incenter
The point where the three angle bisectors of a triangle intersect.
Circumcenter
The point where the three perpendicular bisectors of a triangle intersect.
Orthocenter
The point where the three altitudes of a triangle intersect.
Median
A segment from a vertex of a triangle to the midpoint of the opposite side.
Altitude
A perpendicular segment or line from a vertex of a triangle to the line containing the opposite side.

Common Mistakes to Avoid

  • Confusing medians with perpendicular bisectors is wrong because a median must start at a vertex, while a perpendicular bisector must pass through the midpoint at a 90∘90^\circ angle.
  • Assuming every triangle center is inside the triangle is wrong because the circumcenter and orthocenter can lie outside an obtuse triangle.
  • Using the average of only two vertices to find the centroid is wrong because the centroid uses all three vertices in G(x1+x2+x33,y1+y2+y33)G\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right).
  • Calling the incenter equidistant from the vertices is wrong because the incenter is equidistant from the sides, not from the vertices.
  • Forgetting the centroid's 2:12:1 ratio is wrong because the longer segment is always from the vertex to the centroid, not from the centroid to the midpoint.

Practice Questions

  1. 1 Find the centroid of a triangle with vertices A(2,5)A(2,5), B(8,1)B(8,1), and C(−1,6)C(-1,6).
  2. 2 A median has total length 1818 units. How far is the centroid from the vertex on that median?
  3. 3 A triangle has side lengths 77, 88, and 99 and an inradius of 33. Use A=rsA=rs and s=a+b+c2s=\frac{a+b+c}{2} to find its area.
  4. 4 A triangle is obtuse. Explain which of the four classical centers may lie outside the triangle and why.