Prove that the midpoints of the sides of a quadrilateral lie on a circle if and only if the quadrilateral is orthodiagonal.
For the if part i did as follows:
Given orthodiagonal quadrilateral $ABCD$ we draw diagonals $AC$ and $BD$,from the points of intersection of the diagonals we then draw segment $OP$ where $P$ is the midpoint of $AD$.
Since $\Delta AOD$ is a right triangle we have that $OP$ is its median,hence $OP=AP=PD$.
Now, given that $\Delta APR \cong \Delta OPR$, we see by simmetry that we have $AX=XO$, hence $PX$ is the altitude of isosceles triangle $APO$ ,and from this fact it follows also that $\Delta APX \cong \Delta OPX $.
By the same argument $PY$ is the altitude of isosceles $DPO$,therefore we have $PY=XO$ and $\Delta DPY \cong \Delta OPY \cong \Delta APX \cong \Delta OPX$.
Finally if we consider that $\angle XPO =\angle PDY$ and $\angle OPY =\angle DPY $ we
have that $\angle XPO +\angle OPY =\angle APX +\angle OPY= \angle RPT=90^\circ $.
In this way I can show that quadrilateral $PRST$ is cyclic by showing that also the opposite angle $RST$ is $90^\circ$.
Is this line of reasoning (for sure redundant in some point) okay in general ?In particular can anyone give me some hints on how to approach the only if part ? Also if you can provide any advice on better ways to approach the problem it would be appreciated.
Thanks in advance and forgive me for any english mystakes or if the phrasing wasn't really clear(feel free to edit in case).