I am currently doing a complex number geometry course. I was tasked the following problem:
Problem Statement:
Let P be the point of intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. Further, let M be the midpoint of $AB$. Prove that $AC \perp BD$ if $\overleftrightarrow{PM} \perp CD$ and the center of circe in which $ABCD$ is inscribed doesn't lie on $PM$
I am sorry in advance if my translation to the problem statement is a bit hard to read.
This problem looks very susceptible to complex bashing. But I am not quite sure how to carry out the calculations needed to establish the condition.
We employ complex numbers with the circumcircle of $\triangle ABC$ being the unit circle.
Let
$A = a$, $M = \frac{a+b}{2}$
$B = b$, $P = \frac{ac(b+d) - bd(a+c)}{ac-bd}$
$C = c$
$D = d$
We further have that $\overleftrightarrow{PM} \perp CD$ $\Leftrightarrow$ $pm+cd = 0$ and we want to show that P is the midpoint of one of the diagonals. I am not quite sure how to do that exactly.
Starting Hints ....
(A) Core Equality :
We have ,
(A1) : $ AC⊥BD \iff (c-a)/(d-b) = imaginary \iff (c-a)/(d-b) = - \overline {(c-a)/(d-b)} $
Likewise we have ,
(A2) : $ PM⊥CD \iff (m-p)/(d-c) = imaginary \iff (m-p)/(d-c) = - \overline {(m-p)/(d-c)} $
We have to start with (A2) , plug in the values , then simplify : In Case we get (A1) , we are Done.
(B) Values for ABCD : these are on the Unit Circle , hence $|z|=1$ & $\overline{z}=1/z$
(C) We can take values like $e^{i\theta_1}$ , $e^{i\theta_2}$ , $e^{i\theta_3}$ , $e^{i\theta_4}$ for the 4 Points ABCD.
(D) A Certain Division may involve some term in the Denominator which should not be Zero , hence the Criteria about the Center of the Circle.
I hope this helps ( "... I am not quite sure how to carry out the calculations needed ..." ) & you can Complete it now.