Orthocentre of a Triangle: Interactive Altitudes

Grades 9-10

Question: visualize Orthocentre with slider

In geometry, the orthocentre of a triangle is the unique point where the three altitudes of the triangle intersect. An altitude is defined as a line segment passing through a vertex and perpendicular to the opposite side (or the extension of the opposite side).

Properties of the Orthocentre

As you interact with the slider to move vertex

C

, observe how the position of the orthocentre (

H

) changes relative to the triangle

ABC

Acute-angled Triangle: When all interior angles are less than $90^\circ$ , the orthocentre $H$ lies inside the triangle.
Right-angled Triangle: As vertex $C$ moves such that $\angle ABC = 90^\circ$ or $\angle BAC = 90^\circ$ , the orthocentre $H$ will coincide exactly with the vertex of the right angle.
Obtuse-angled Triangle: When one interior angle is greater than $90^\circ$ , the altitudes from the acute angles must be extended outside the triangle to meet. Consequently, the orthocentre $H$ lies outside the triangle.

Geometric Visualization

Altitudes (Dashed Lines): These lines pass through each vertex ( $A, B, C$ ) and are perpendicular to the opposite sides.
Right-Angle Markers: The small squares at the feet of the altitudes indicate that the lines are perpendicular [altitude $\perp$ opposite side].
Orthocentre ( $H$ ): The point where all three altitudes meet.

Note that for any triangle, the orthocentre, circumcentre, and centroid are collinear, forming a line known as the Euler line (except in equilateral triangles where they coincide).