synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. examples of such proofs can be found in the articles butterfly theorem, angle bisector theorem, apollonius' theorem, british flag theorem, ceva's theorem, equal incircles theorem, geometric mean theorem, heron's formula, isosceles triangle theorem, law of cosines, and others that are linked to .