Page:First six books of the elements of Euclid 1847 Byrne.djvu/69

HE opposite sides and angles of any parallelogram are equal, and the diagonal () divides it into two equal parts.

Since $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	=	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (pr. 29.)
	=

and common to the two triangles.

∴ $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	=	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (pr. 26.)
	=
	=

and = (ax.):

Therefore the opposite sides and angles of the parallelogram are equal: and as the triangles and are equal in every respect (pr. 4.), the diagonal divides the parallelogram into two equal parts.

Q. E. D.