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 2 of 6: Reflect the tangent line across the three sides
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In plain words

Reflecting two convenient points of the tangent line across a side (using the standard mirror formula for chords of the unit circle) and threading a line through their images produces a clean closed-form equation for each reflected line.

t2z+(bc)2zˉ=(b+c)(t2+bc)−2t bct^2z+(bc)^2\bar z=(b+c)(t^2+bc)-2t\,bc
Detailed analysis

Reflection across the chord through unit points p,qp,q sends a point zz to p+q−pqzˉp+q-pq\bar z. Applying this to two convenient points of ℓ\ell (the tangency point tt itself, and one other point of ℓ\ell) across side BCBC (through b,cb,c) and threading the complex-line equation through their images gives, after simplification, the reflected line ℓa: t2z+(bc)2zˉ=(b+c)(t2+bc)−2t bc\ell_a:\ t^2z+(bc)^2\bar z=(b+c)(t^2+bc)-2t\,bc. The same computation across CACA and ABAB gives ℓb: t2z+(ca)2zˉ=(c+a)(t2+ca)−2t ca\ell_b:\ t^2z+(ca)^2\bar z=(c+a)(t^2+ca)-2t\,ca and ℓc: t2z+(ab)2zˉ=(a+b)(t2+ab)−2t ab\ell_c:\ t^2z+(ab)^2\bar z=(a+b)(t^2+ab)-2t\,ab.