MathLabs

Problem 6

Let ABCABC be an acute triangle with circumcircle Γ\Gamma. Let ℓ\ell be a tangent line to Γ\Gamma, and let ℓa\ell_a, ℓb\ell_b and ℓc\ell_c be the lines obtained by reflecting ℓ\ell in the lines BCBC, CACA and ABAB, respectively. Show that the circumcircle of the triangle determined by the lines ℓa,ℓb,ℓc\ell_a,\ell_b,\ell_c is tangent to the circle Γ\Gamma.
Step 1 of 6: Turn the picture into complex numbers
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In plain words

Placing the circumcircle as the unit circle turns every vertex into a unit complex number, and there is a ready-made formula for the tangent line at any point of that circle.

z+t2zˉ=2tz+t^2\bar z=2t
Detailed analysis

Place Γ\Gamma as the unit circle in the complex plane, and let a,b,c,ta,b,c,t be the unit complex numbers representing A,B,CA,B,C and the point of tangency TT of ℓ\ell. A standard fact about the unit circle is that the tangent line at a point tt on it has equation z+t2zˉ=2tz+t^2\bar z=2t: indeed z=tz=t satisfies it since t+t2⋅(1/t)=2tt+t^2\cdot(1/t)=2t, and the coefficient t2t^2 (of unit modulus) certifies it really is a line rather than a single point.