33. The Altitude's Path · meeting at the orthocentre

Try to break it

Drag A, B, or C. Watch how the altitudes stretch and tilt, but they always stay perpendicular to their sides.

How you build it

Construct the three altitudes of a triangle.

• Place vertex A of the triangle.
• Place vertex B.
• Place vertex C, clearly off the line AB so the triangle has real area.
• Draw side AB (Segment tool: click A, then B).
• Draw side BC (Segment tool: click B, then C).
• Draw side CA (Segment tool: click C, then A).
• Altitude from A — perpendicular to the opposite side BC. With the Perp tool, click vertex A, then click on side BC. The tool drops a perpendicular from A straight onto BC.
• Altitude from B — perpendicular to the opposite side CA. With the Perp tool, click vertex B, then click on side CA.
• Altitude from C — perpendicular to the opposite side AB. With the Perp tool, click vertex C, then click on side AB. The three altitudes now cross at one point — the orthocentre.

The proof, step by step

Prove that the three altitudes of the triangle meet at a single point.

• An altitude of a triangle is the perpendicular segment drawn from a vertex to its opposite side.
• A triangle has three vertices, so three altitudes can be drawn — one from each vertex.
• When all three altitudes are drawn, they cross one another at a single common point.
• That single point where the three altitudes meet is called the orthocentre.

Worked example

In a triangle, the point where all three altitudes meet is called the:

The point of concurrency of the three altitudes of a triangle is known as the orthocentre.

• Centroid
• Orthocentre — correct
• Circumcentre
• Incentre