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Point $A,B,C$, all different, lies in unit circle on complex plane. $X$ is midpoint of $BC$. $AX$ cuts the circle again at point $Y \neq A$. Express $y$ in terms of $a,b,c$. (Point $A$ is denoted by the complex number $a$, and similar notation for others)

My work: Since $A,X,Y$ are collinear, so $y=aw+x(1-w)$, for a real number $w$. Since $Y$ lies on the circle, $y\overline{y}=1$ and similar with others. I can use $y\overline{y}=1$ to find $w$ and note that $X=\frac{b+c}{2}$.

My way is too long and requires solving a quadratic equation. Any better way?

Invariance
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Adola
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1 Answers1

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In https://studymath.github.io/assets/docs/real_complex_bash.pdf we read:

If $AB$ is a chord of the unit circle, the equation of line $AB$ is given by $$z = a + b − ab\overline{z}.$$

In the present question is the chord $AY,$ the equation is $$z = a + y − ay\overline{z}.$$ The midpoint of $BC$ lies on $AY$ says $$\frac{b+c}{2}=a+y-ay\frac{\overline{b+c}}{2},$$ from where $$y=\frac{b+c-2a}{2-a\cdot{\overline{b+c}}}$$

user376343
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