Proof that All Triangles can be Inscribed into a Circle

Question

Fellow Mathers, I am currently in a Euclidean Geometry class. We are using the SMSG axioms to build our geometry. I am struggling in proving the fact that all triangles can be inscribed into a circle, i.e. that...

$Ɐ$ triangles $ΔABC$, $∃$ a circle $Θ$ such that $A,B,C ∈ Θ$.

Anyone who can provide a proof or point me in the right direction, it would be greatly appreciated. - SDH

Answer 1

Let be $D$ the intersection point of the perpendicular bisectors of $AC$ and $BC$. You have that

1. $AD=CD$ since $D$ belongs to $s$
2. $CD=BD$ since $D$ belongs to $r$ Therefore $AD=BD=CD=R $ and the circle of center $D$ and radius $R$ is the required circle.