Abstract In his paper Über die Anwendung der
komplexen Zahlen in der Elementargeometrie}, Bilten na
Društvoto na matematičarite i fizičarite od N.R.
Makedonija, kn. I, Skopje, (1950), 5581 (in Serbian, resume in
German) and in the university textbook Complex Numbers, Belgrade,
1950, Jovan Karamata (19021967) maintains that, in planimetry, a
complex number can assume the role of a vector, whereby in
definition and solving of certain problems, there appears the
product $\overline{a} b=A+Bi$ where $\overline{a}$ is the conjugated
number of $a$, while $a$ and $b$ are complex numbers which
correspond to free vectors $\vec{a}$ and $\vec{b}$. Using
geometric interpretation of $a$ and $b$, Karamata expresses $A$
and $B$ as $A=ab\cos \alpha$ and $B=ab\sin
\alpha$. In order to underline the geometric sense of these
expressions, Karamata denotes them as $A=(a\perp b)$, resp.
$B=(a\mid b)$ designating them ``orthogonal
product'' and ``parallel product''. By means of these two symbols,
considered as products, Karamata interprets those problems in
planimetry which correspond to parallelity and orthogonality, and
shows how they can be used in deriving of PaposPascal and
Desargues propositions.

Keywords: Karamata, projective geometry, complex number,
vector, PaposPascal, Desargues. 