Circumcentre of a Triangle: Interactive Perpendicular Bisectors

Grades 9-10
Interactive Circumcentre of a Triangle: Interactive Perpendicular Bisectors graph — click to explore
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Question: visualize Circumcentre with slider
In the study of coordinate geometry and the properties of triangles in the HKDSE, the circumcentre is one of the four centers of a triangle that students must master. It is defined as the point of intersection of the perpendicular bisectors of the three sides of a triangle.

1. Mathematical Properties

The circumcentre (often denoted as O) has several unique properties:
  • Equidistance: The circumcentre is equidistant from all three vertices of the triangle. That is, OA = OB = OC = R, where R is the circumradius.
  • The Circumcircle: Because it is equidistant from the vertices, it serves as the centre of a circle that passes through A, B, and C, known as the circumcircle.
  • Geometric Construction: To find the circumcentre, we construct the perpendicular bisectors of at least two sides. The point where they meet is the circumcentre [\perp \text{ bisector of chord passes through centre}].

2. Position Relative to the Triangle

The position of the circumcentre depends entirely on the type of triangle:
  • Acute Triangle: The circumcentre lies inside the triangle.
  • Right-angled Triangle: The circumcentre lies exactly on the midpoint of the hypotenuse. This is consistent with the theorem that the angle in a semi-circle is a right angle [\angle \text{ in semi-circle}].
  • Obtuse Triangle: The circumcentre lies outside the triangle, specifically on the opposite side of the longest side from the obtuse angle.

3. Visualization Guide

In the following diagram:
  • Vertices A and B are fixed on the x-axis.
  • Vertex C moves horizontally via the slider k.
  • The blue dashed lines represent the perpendicular bisectors of the sides.
  • The red point O is the circumcentre.
  • The orange circle is the circumcircle.
By moving the slider, you can observe the circumcentre shifting. Notice how it "exits" the triangle when the vertex C moves far enough to create an obtuse angle at A or B. When the triangle is right-angled (e.g., when C is directly above A), the circumcentre sits perfectly on the midpoint of the side BC.
助提示 (DSE Tip): In Paper 1 long questions, if you are asked to find the coordinates of the circumcentre, you should either:
  1. Equate the distances: \sqrt{(x-x_1)^2 + (y-y_1)^2} = \sqrt{(x-x_2)^2 + (y-y_2)^2}.
  2. Find the equations of two perpendicular bisectors and solve them as simultaneous equations.
要求: [ \perp bisector of chord passes through centre]
geometrytriangle-centers