Projection set (triangle)

Equal area of ​​the two gray rectangles:
with and applies ${\displaystyle c=AB,\,b=AC}$
${\displaystyle p_{bc}=AE,\,p_{cb}=AD}$
${\displaystyle |c|\cdot |p_{bc}|=|b|\cdot |p_{cb}|}$

The projection theorem is a statement from elementary geometry , which represents a generalization of the cathetus theorem to any triangle :

For two sides in any triangle, those rectangles are of equal area which are formed from one side and the orthogonal projection of the other onto it.

In any triangle with sides a , b and c let p _xy denote the projection of side x onto side y , then: ${\displaystyle \triangle ABC}$

${\displaystyle {\begin{aligned}|c|\cdot |p_{bc}|&=|b|\cdot |p_{cb}|\\|a|\cdot |p_{ba}|&=|b|\cdot |p_{ab}|\\|c|\cdot |p_{ac}|&=|a|\cdot |p_{ca}|\\\end{aligned}}}$

In a right triangle with a right angle in C , the projections p _cb and p _ca correspond to sides b and a . The projection theorem then delivers: ${\displaystyle \triangle ABC}$

${\displaystyle {\begin{aligned}|c|\cdot |p_{bc}|&=|b|^{2}\\|c|\cdot |p_{ac}|&=|a|^{2}\end{aligned}}}$

So one obtains the Euclidean prophecy .

literature

• Hans Schupp: Elementarge Geometry ( Uni pocket books 669 Mathematics ). Schöningh, Paderborn 1977, ISBN 3-506-99189-2 , pp. 117-118