Centroid, Incentre, Circumcentre and Orthocentre of a Right Triangle

Grades 9-10
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Question: Four centers in a right angled triangle
In a right-angled triangle, the positions of the four classical triangle centers—the Orthocenter, Circumcenter, Centroid, and Incenter—exhibit unique properties and specific geometric relationships.

1. The Orthocenter ( H)

The orthocenter is the point where the three altitudes of a triangle intersect. In a right-angled triangle, two of the altitudes are actually the legs of the triangle itself.
  • Property: The orthocenter always coincides with the vertex containing the right angle.
  • In the diagram, observe how Point H (red) stays fixed at the origin if the legs are aligned with the axes.

2. The Circumcenter ( O)

The circumcenter is the point where the perpendicular bisectors of the sides meet, and it serves as the center of the circle passing through all three vertices (the circumcircle).
  • Property: In a right-angled triangle, the circumcenter is exactly the midpoint of the hypotenuse.
  • The distance from O (blue) to any vertex is equal to half the length of the hypotenuse ( R = \frac{c}{2}).

3. The Centroid ( G)

The centroid is the intersection of the three medians (lines connecting vertices to the midpoints of opposite sides). It represents the geometric center of mass.
  • Property: The centroid G (green) is always located inside the triangle. In a coordinate system where the right angle is at (0,0), its coordinates are the average of the vertices: (\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}).

4. The Incenter ( I)

The incenter is the intersection of the angle bisectors and is the center of the inscribed circle (incircle).
  • Property: The incenter I (orange) is always inside the triangle. For a right triangle with legs a and b and hypotenuse c, the radius of the incircle is r = \frac{a + b - c}{2}.

The Euler Line

In any triangle (except equilateral ones), the Orthocenter ( H), Centroid ( G), and Circumcenter ( O) are collinear. This line is known as the Euler Line.
  • In a right-angled triangle, the Euler line connects the right-angle vertex to the midpoint of the hypotenuse.
  • Notably, the distance HG is always exactly twice the distance GO ( HG = 2 \cdot GO).
  • The Incenter ( I) generally does not lie on the Euler line unless the triangle is isosceles.
You can use the sliders in the diagram to change the lengths of the triangle's legs and see how these four centers move and interact.
geometrytriangle-centers