Geometry • Topic 11

Orthocenter

The orthocenter ( $H$ ) is the intersection of the three altitudes. While it doesn't have a direct "circle" like $O$ or $I$ , it has rich symmetric properties.

Definition

Definition (Orthocenter)

The orthocenter is the concurrent point of the three altitudes (perpendiculars from vertices to opposite sides).

Properties

1. Angles

The angles formed at the orthocenter are supplementary to the vertex angles:

∠ B H C = 18 0^{\circ} - A

(Note: This means

∠ B H C + ∠ A = 18 0^{\circ}

, so

A B H C

is not quite cyclic, but related).

2. Reflections

The reflection of $H$ across any side lies on the circumcircle.
The reflection of $H$ across the midpoint of any side lies on the circumcircle (specifically, at the point diametrically opposite the vertex).

3. Orthic Triangle

The triangle formed by the feet of the altitudes

D, E, F

is the orthic triangle.

$H$ is the incenter of the orthic triangle (if $A BC$ is acute).

Power of a Point

A D, BE, CF

are altitudes:

A H \cdot HD = B H \cdot H E = C H \cdot H F

(This power relates to the circle with diameter

A B

, etc.).

Practice Problems

Exercise (Problem 1)

H

is the orthocenter of

△ A BC

, prove that

A

is the orthocenter of

△ H BC

Exercise (Problem 2)

In acute

△ A BC

∠ A = 4 5^{\circ}

and

BC = 10

. Find the length of

A H

Exercise (Problem 3)

Prove that the distance from the circumcenter to a side equals half the distance from the orthocenter to the opposite vertex (

A H = 2 O M_{a}