Add extra expressions which match exp term of d bound

main.tex

@@ -246,6 +246,15 @@ This can be rearranged to express a bound on $d$ as follows:
 		+ C*q
 		- q^2
 	).expand()/(R-2*r)
+	foo = (
+		- 2*D*R + (C^2)/4
+		- ((C - 4*R*beta_min)/2 - q)^2
+	).expand()/(R-2*r)
+	foo = (
+		(2*R*beta_min + q)
+		*(2*R*beta_min + q - C)
+		+ 2*D*R
+	).expand()/(2*r - R)
 \end{sagesilent}
 
 \begin{equation}
@@ -322,6 +331,7 @@ This can be rearranged to express a bound on $d$ as follows:
 
 \section{Conclusion}
 
+\newpage
 \section{Appendix - SageMath code}
 
 \definecolor{codegreen}{rgb}{0,0.6,0}