Affine Räume/Produktraum/Punktbeziehung/Aufgabe/Lösung

Zur Auswertung des baryzentrischen Ausdruckes wählen wir ${}(P_{1},Q_{1})$ als Aufpunkt und erhalten

{}{\begin{aligned}(P_{1},Q_{2})+(P_{2},Q_{1})-(P_{1},Q_{1})&=(P_{1},Q_{1})+{\overrightarrow {(P_{1},Q_{1})(P_{1},Q_{2})}}+{\overrightarrow {(P_{1},Q_{1})(P_{2},Q_{1})}}\\&=(P_{1},Q_{1})+({\overrightarrow {P_{1}P_{1}}},{\overrightarrow {Q_{1}Q_{2}}})+({\overrightarrow {P_{1}P_{2}}},{\overrightarrow {Q_{1}Q_{1}}})\\&=(P_{1},Q_{1})+(0,{\overrightarrow {Q_{1}Q_{2}}})+({\overrightarrow {P_{1}P_{2}}},0)\\&=(P_{1}+{\overrightarrow {P_{1}P_{2}}},Q_{1}+{\overrightarrow {Q_{1}Q_{2}}})\\&=(P_{2},Q_{2}).\end{aligned}}