Can we use Symmetry as valid proof in geometry

Question

My question is that can we use the symmetry of a shape as an valid proof in geometry? for example, according to following shape, we have an Isosceles trapezoid and we draw its inner bisectors. we want to prove that the shape made from that is an kite. (in shape $MQ = MN$ AND $NP = NQ$) Is it OK to say that because Isosceles trapezoid is a symmetrical shape, so there is no difference between the bisector of an angle from one side to the other and so, the shape made from this will also have this symmetry? In shape ABCD is an Isosceles trapezoid and we want to prove MNPQ is a kite.

Sorry because not-well-drawn shape and my English.

Thanks.

Answer 1

Sure. Formally, let $\ell$ be the bisector of $CD$. Then reflection at $\ell$ maps $C\mapsto D$, by equality of angles maps line $CB$ to $AD$ and line $AB$ to itself because it is perpendicular to $\ell$, hence maps $B\mapsto D$. It also maps line $CM$ to $DM$ and line $BP$ to line $AP$. We conclude $M,P\in \ell$ and $N\mapsto Q$, so that $MNPQ$ is a quadrilateral whose one diagonal is a symmetry axis.

Answer 2

This is not an answer to your question but is a response to your title.

Through my high school study of Geometry, I used the text called “A Concise Geometry” by Clement V. Durell.

After some searching through that book, I found:-

1. Neither the terms “symmetry” nor “axis of symmetry” can be found in its glossary and index.

2. They can be found in some the question statements but have not been properly introduced in any of its earlier parts.

This could mean that the term is assumed to be commonly known / accepted and its properties should also be well understood. However, the idea symmetry has not been applied as proofs in any of the subsequently developed theorems or reasons in any of the following worked examples.

Later, we found its logic is sound and is in fact an excellent tool in proving geometric problems. We then simply use it but we don’t know who is the starter of introducing it formally.