Question

(a) Suppose $x$ and $y$ are points on the unit circle such that the line through $x$ and $y$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = \dfrac{x+y}{xy+1}$.

(b) Let $Q_1 Q_2 \dotsb Q_{18}$ be a regular 18-gon. Show that $Q_1 Q_{10}$, $Q_2 Q_{13}$, and $Q_3 Q_{15}$ are concurrent.

Answer 1

Not sure how you can call x and y points, and then use them as values in an expression.

Most precal courses don't cover vectors, and even so, the expression is not a valid vector computation.

I guess you could use complex numbers for the points, and then they would be

x = √(1-a^2) + ai
y = √(1-b^2) + bi
where a≠b

Then the line through the points would be

y = (b-a)/(√(1-b^2) - √(1-a^2)) (x-√(1-a^2)) + b

You could show that the x-intercept of that line would correspond the the given expression.

But maybe I'm misinterpreting the situation.