In projective geometry, the **harmonic conjugate point** of a point on the real projective line with respect to two other points is defined by the following construction:

- Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively. If AN and BM meet at K, and LK meets AB at D, then D is called the
**harmonic conjugate**of C with respect to A and B.^{[1]}

The point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N. This fact follows from Desargues theorem.

In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as (*A*, *B*; *C*, *D*) = −1.

The four points are sometimes called a harmonic range (on the real projective line) as it is found that D always divides the segment AB *internally* in the same proportion as C divides AB *externally*. That is:

If these segments are now endowed with the ordinary metric interpretation of real numbers they will be *signed* and form a double proportion known as the cross ratio (sometimes *double ratio*)

for which a harmonic range is characterized by a value of −1. We therefore write:

The value of a cross ratio in general is not unique, as it depends on the order of selection of segments (and there are six such selections possible). But for a harmonic range in particular there are just three values of cross ratio: {−1, 1/2, 2}, since −1 is self-inverse – so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the **harmonic cross-ratio**.

In terms of a double ratio, given points a, b on an affine line, the **division ratio**^{[2]} of a point x is
Note that when *a* < *x* < *b*, then *t*(*x*) is negative, and that it is positive outside of the interval.
The cross-ratio is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when *t*(*c*) + *t*(*d*) = 0, then c and d are harmonic conjugates with respect to a and b. So the division ratio criterion is that they be additive inverses.

Harmonic division of a line segment is a special case of Apollonius' definition of the circle.

In some school studies the configuration of a harmonic range is called *harmonic division*.

When x is the midpoint of the segment from a to b, then
By the cross-ratio criterion, the harmonic conjugate of x will be y when *t*(*y*) = 1. But there is no finite solution for y on the line through a and b. Nevertheless,
thus motivating inclusion of a point at infinity in the projective line. This point at infinity serves as the harmonic conjugate of the midpoint x.

Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as KLMN in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:

- D is the harmonic conjugate of C with respect to A and B, which means that there is a quadrangle IJKL such that one pair of opposite sides intersect at A, and a second pair at B, while the third pair meet AB at C and D.
^{[3]}

It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:

- ...Staudt succeeded in freeing projective geometry from elementary geometry. In his
*Geometrie der Lage*, Staudt introduced a harmonic quadruple of elements independently of the concept of the cross ratio following a purely projective route, using a complete quadrangle or quadrilateral.^{[4]}

To see the complete quadrangle applied to obtaining the midpoint, consider the following passage from J. W. Young:

- If two arbitrary lines AQ, AS are drawn through A and lines BS, BQ are drawn through B parallel to AQ, AS respectively, the lines AQ, SB meet, by definition, in a point R at infinity, while AS, QB meet by definition in a point P at infinity. The complete quadrilateral PQRS then has two diagonal points at A and B, while the remaining pair of opposite sides pass through M and the point at infinity on AB. The point M is then by construction the harmonic conjugate of the point at infinity on AB with respect to A and B. On the other hand, that M is the midpoint of the segment AB follows from the familiar proposition that the diagonals of a parallelogram (PQRS) bisect each other.
^{[5]}

Four ordered points on a projective range are called **harmonic points** when there is a tetrastigm in the plane such that the first and third are codots and the other two points are on the connectors of the third codot.^{[6]}

If p is a point not on a straight with harmonic points, the joins of p with the points are **harmonic straights**. Similarly, if the axis of a pencil of planes is skew to a straight with harmonic points, the planes on the points are **harmonic planes**.^{[6]}

A set of four in such a relation has been called a **harmonic quadruple**.^{[7]}

A conic in the projective plane is a curve C that has the following property:
If P is a point not on C, and if a variable line through P meets C at points A and B, then the variable harmonic conjugate of P with respect to A and B traces out a line. The point P is called the **pole** of that line of harmonic conjugates, and this line is called the **polar line** of P with respect to the conic. See the article Pole and polar for more details.

Main article: Inversive geometry |

In the case where the conic is a circle, on the extended diameters of the circle, harmonic conjugates with respect to the circle are inverses in a circle. This fact follows from one of Smogorzhevsky's theorems:^{[8]}

- If circles k and q are mutually orthogonal, then a straight line passing through the center of k and intersecting q, does so at points symmetrical with respect to k.

That is, if the line is an extended diameter of k, then the intersections with q are harmonic conjugates.

Consider as the curve an ellipse given by the equation

Let be a point outside the ellipse and a straight line from which meets the ellipse at points and . Let have coordinates . Next take a point on and inside the ellipse which is such that divides the line segment in the ratio to , i.e.

- .

Instad of solving these equations for and it is easier to verify by substitution that the following expressions are the solutions, i.e.

Since the point lies on the ellipse , one has

or

This equation - which is a quadratic in - is called Joachimthal's equation. Its two roots , determine the positions of and in relation to and . Let us associate with and with . Then the various line segments are given by

and

It follows that

When this expression is , we have

Thus divides ``internally´´ in the same proportion as divides ``externally´´. The expression

with value (which makes it self-inverse) is known as the harmonic cross ratio. With as above, one has and hence the coefficient of in Joachimthal's equation vanishes, i.e.

This is the equation of a straight line called the polar (line) of point (pole) . One can show that this polar of is the chord of contact of the tangents to the ellipse from . If we put on the ellipse () the equation is that of the tangent at . One can also sho that the directrix of the ellipse is the polar of the focus.

In Galois geometry over a Galois field GF(*q*) a line has *q* + 1 points, where ∞ = (1,0). In this line four points form a harmonic tetrad when two harmonically separate the others. The condition

characterizes harmonic tetrads. Attention to these tetrads led Jean Dieudonné to his delineation of some accidental isomorphisms of the projective linear groups PGL(2, *q*) for *q* = 5, 7, 9.^{[9]}

If *q* = 2^{n}, and given A and B, then the harmonic conjugate of C is itself.^{[10]}

Let *P*_{0}, *P*_{1}, *P*_{2} be three different points on the real projective line. Consider the infinite sequence of points P_{n}, where P_{n} is the harmonic conjugate of *P*_{n-3} with respect to *P*_{n-1}, *P*_{n-2} for *n* > 2. This sequence is convergent.^{[11]}

For a finite limit P we have

where is the golden ratio, i.e. for large n. For an infinite limit we have

For a proof consider the projective isomorphism

with