If in a triangle $ABC$, $A \equiv (1,10)$, circumcenter $\equiv (-\frac13, \frac23)$ and orthocenter $\equiv (\frac{11}3, \frac43)$ then the coordinates of mid-point of side opposite to A is?
Here clearly point $Q$ is circumcenter and point $P$ is orthocenter. The only thing I see here is $AP ||DQ$. So their slopes are same. So we can get an equation using this. But how am I supposed to get another equation?
Any other way of solving this problem would also be appreciated.
Hint: $~$ You were given the orthocenter H and the circumcenter O. But what were you not
given ? The centroid G. Now, do you, by any chance, know that the latter is found on the
segment uniting the former two, and, not only that, but it divides said segment in a ratio of
$1:2$ ? $($Remember that the centroid also divides each median into the same ratio as well$).$
In other words, $G\in(OH)$ and $GH=2GO.~$ At the same time, $G=\dfrac{A+B+C}3,~$ and
$M=\dfrac{B+C}2.$