Let P,Q ,R be points represented by complex numbers ` z_(1),z_(2),z_(3)` and circumcentre of ` DeltaPQR` conicides with origin, Let the altitude , PL of the ` Delta` meets the cricumircle again at M, then find the complex number representing the point M.

Explanation

form figure, ltBrgt |OP| = `|z_(1)|`
` |OQ|= |z_(2)|`
`|OR|= |z_(3)|`
and | OM| = |z|
let ` angleMPR= theta , "then" angleMOR = 2theta `
Also ,` angle PLR = pi/2, pi/2 , anglePRQ = pi/2 - theta`
` anglePOQ = 2(pi/2 - theta) = pi-2theta`
Rotating M about O through an angle ` 2 theta` in anticlockwise direction, we have ,
` z_(3)/z= (|z_(3)|)/(|z|)e^(2itheta)`
Also , `z_(2)/z_(1)= (|z_(2)|)/|z_(1)|e^(i(pi-2theta))= - e^(-2itheta), " as " |z_(1)| = |z_(2)| = |z_(3)|= |z|` ( Rotating P about O in anti-clockwise direction).
`Rightarrow z_(3)/z= z_(2)/z_(1) = |z_(3)|/|z| . |z_(2)|/|z_(1)|(-1) = -1`
` Rightarrow z= - (z_(2)z_(3))/z_(1)`
Thus the point M is determined by the complex number ` - (z_(2)z_(3))/z_(1)`