For a triangle with sides , three straight lines are given, each of which is perpendicular to an (extended) side of the triangle and which intersect at a common point . Denoting the base points on the (extended) triangle sides with , then the following equation holds:
.
The reverse of this proposition also applies, that is: If the base points of three perpendiculars meet the above equation, then they intersect at a common point.
Special cases
If the triangle has a right angle and the point of intersection lies on one of the two corner points or , one obtains the Pythagorean theorem. Is, for example, on , then applies , , , , and , and supplies the above equation .
If the three straight lines are perpendicular to the center , then , and . Hence the above equation exists and as a special case we get the theorem that the perpendiculars of a triangle intersect at a point.
If the three straight lines are the extensions of the triangular heights , the straight lines run through the corner points. The height divides the triangle into two right-angled triangles, for which the Pythagorean theorem provides the equations and , and follows by taking the difference . The relationships and follow in exactly the same way or by mentally rotating the triangle . If you add these three relationships, you get
,
that is, the equation consists of the above sentence. So we also get the theorem of the vertical intersection as a special case of Carnot's theorem.
literature
Martin Wohlgemuth (Ed.): Mathematical for advanced beginners. More popular posts from Matroids Matheplanet. Springer, 2010, ISBN, 9783827426079, pp. 273-276.