Harcourt's theorem is a formula in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

for the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...

of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

.. The theorem is named after J. Harcourt, an Irish professor..

Statement

Let a

be given with vertices ''A'', ''B'', and ''C'', opposite sides of lengths ''a'', ''b'', and ''c'', area ''K'', and a line that is

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

to the triangle's

at any point on that circle. Denote the signed perpendicular distances of the vertices from the line as ''a'' ', ''b'' ', and ''c'' ', with a distance being negative if and only if the vertex is on the opposite side of the line from the incenter. Then :

a a ^\prime + b b^\prime + c c^\prime = 2K.

Degenerate case

If the tangent line contains one of the sides of the triangle, then two of the distances are zero and the formula collapses to the familiar formula that twice the area of a triangle is a base (the coinciding triangle side) times the altitude from that base.

Extension

If the line is instead tangent to the

excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

opposite, say, vertex ''A'' of the triangle, then :

-a a ^\prime + b b^\prime + c c^\prime = 2K.

Dual property

If rather than ''a', b', c' '' referring to distances from a vertex to an arbitrary incircle tangent line, they refer instead to distances from a sideline to an arbitrary point, then the equation :

a a ^\prime + b b^\prime + c c^\prime = 2K.

remains true.Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books

References

{{reflist Theorems about triangles and circles