Theorem

The orthocentre and the orthocentric system

Geometry • Triangle • Triangle Centres

Statement

The three altitudes of a triangle are concurrent, and their common point is the orthocentre. If

H

is the orthocentre of

△ A B C

, then the four points

A

B

C

H

form an orthocentric system: each one is the orthocentre of the triangle formed by the other three, so

A

is the orthocentre of

△ H B C

B

△ H C A

and

C

△ H A B

How to use it

1To find an altitude of one of these triangles, reuse a perpendicular already drawn: the line through $H$ perpendicular to a side is an altitude of the original triangle and of the new one alike.
2Swap a vertex for the orthocentre and the configuration is unchanged, which turns four separate questions into one.
3In a right-angled triangle the orthocentre is the vertex carrying the right angle, and in an obtuse triangle it lies outside the triangle.

Worked example

H

is the orthocentre of

△ A B C

. Of which triangle is

C

the orthocentre?

Step-by-step

1
$A, B, C, H$ form an orthocentric system
Why: $H$ is the orthocentre of the triangle on the other three points.
2
$C$ is the orthocentre of $△ H A B$
Why: Exchange $C$ with $H$ and take the triangle on the remaining three points.

Answer

△ H A B

Watch out

The exchange pairs a point with the triangle on the OTHER three, so the point must not be a vertex of that triangle; and the orthocentre is not the centre of any circle through the vertices, so it must not be confused with the circumcentre or the incentre.

Builds on

Angles on a straight line and vertically opposite angles

Also known as

orthocentre, orthocenter, altitudes are concurrent, orthocentric system, point of concurrence of altitudes

Appears in past papers

Statement

Worked example

Appears in past papers

2019 (II)

2018 (I)

2017 (II)