Question

Let the line segments AB and CD intersect at O in such a way that OA = OD and OB = OC. Prove that AC = BD but AC may not be parallel to BD.

Answer 1

$$\begin{gathered} \mathrm{In}△\mathrm{OAC}\mathrm{and}△\mathrm{ODB},\mathrm{we}\mathrm{get}: \\ \mathrm{OA}=\mathrm{OD}\left(\mathrm{Given}\right) \\ \mathrm{OC}=\mathrm{OB}\left(\mathrm{Given}\right) \\ \angle \mathrm{AOC}=\angle \mathrm{BOD}(\mathrm{Vertically}\mathrm{opposite}\mathrm{angle}) \\ △\mathrm{OAC}\cong △\mathrm{ODB}(\mathrm{SAS}\mathrm{criterion}) \end{gathered}$$
$\mathrm{i}.\mathrm{e}.,\mathrm{AC}=\mathrm{BD}\left(\mathrm{CPCT}\right)$

However, $\angle OAC$ is not equal to $\angle OBD$ and $\angle OCA$ is not equal to $\angle ODB$.
Thus, AC may not be parallel to BD.
Hence, proved.