2 Areas of Corresponding Triangles.

Article 2
Areas of Corresponding Triangles.

The areas of corresponding triangles have equal measures. For, let the coordinates of $P_{1}, Q_{1}$ be $(x_{1}, y_{1}), (x'_{1}, y'_{1})$ , and let those of their correspondents $P_{2}, Q_{2}$ be $(x_{2}, y_{2}), (x'_{2}, y'_{2})$ ; let the triangles $P_{1} O_{1} Q_{1}, P_{2} O_{2} Q_{2}$ be $T_{1}, T_{2}$ , and let the measuring triangles $A_{1} O_{1} B_{1}, A_{2} O_{2} B_{2}$ be $K_{1}, K_{2}$ , and their angles $ω_{1}, ω_{1}$ ; then, by analytic geometry, taking account of both magnitude and direction of angles, areas, and lines,

\begin{matrix} \begin{aligned} \frac{T_{1}}{K_{1}} & = \frac{\frac{1}{2} (x_{1} y'_{1} - x'_{1} y_{1}) sin ω_{1}}{\frac{1}{2} a_{1} b_{1} sin ω_{1}} = \frac{x_{1}}{a_{1}} \cdot \frac{y'_{1}}{b_{1}} - \frac{x'_{1}}{a_{1}} \cdot \frac{y_{1}}{b_{1}}; \\ \frac{T_{2}}{K_{2}} & = \frac{\frac{1}{2} (x_{2} y'_{2} - x'_{2} y_{2}) sin ω_{2}}{\frac{1}{2} a_{2} b_{2} sin ω_{2}} = \frac{x_{2}}{a_{2}} \cdot \frac{y'_{2}}{b_{2}} - \frac{x'_{2}}{a_{2}} \cdot \frac{y_{2}}{b_{2}} . \end{aligned} & T & h & e & r & e & f & o & r & e & , & b & y & ( & 2 & ) & , \\ \frac{T_{1}}{K_{1}} = \frac{T_{2}}{K_{2}} . & (3) \end{matrix}

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Article 2Areas of Corresponding Triangles.

Article 2
Areas of Corresponding Triangles.