Harcourt's theorem

Harcourt's theorem with positive (blue) and negative (red) distances a ', b' and c ': ${\displaystyle 2F=aa'+bb'+cc'=a|a'|+b|b'|-c|c'|}$

The set of Harcourt is a statement in the elementary geometry, a relationship between the area of a triangle and the intervals of its corner points of a tangent of its inscribed circle describes.

For any triangle ABC with side lengths a , b and c, let a ' , b' and c 'be the signed distances of the corner points A , B and C from a tangent of the inscribed circle. Then it holds that the sum of the products of signed distance and side length corresponds to twice the area of ​​the triangle:

${\displaystyle 2\cdot F=a\cdot a'+b\cdot b'+c\cdot c'}$

Here, the distance has a positive sign if it is on the same side of the tangent as the inscribed circle, and a negative sign if it is on the other side of the tangent.

The set is named after the Irish math professor J. Harcourt (around 1900).

Special case

If the tangent coincides with the straight line through and , then with the above notation and is the height . Harcourt's theorem in this case is or . This is the well-known formula for the area of ​​a triangle . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle a'=b'=0}$ ${\displaystyle c'}$ ${\displaystyle h_{c}}$ ${\displaystyle 2\cdot F=c\cdot h_{c}}$ ${\displaystyle \textstyle F={\frac {1}{2}}c\cdot h_{c}}$

literature

• Nikolaos Dergiades, Juan Carlos Salazar: Harcourt's Theorem . Forum Geometricorum, Volume 3, 2003, pp. 117-124.
• G. Marie: Exercices de géométrie . 5th edition, 1912, p. 750