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 5 of 6: Conjugate using that a, b, c, t sit on the unit circle
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In plain words

Since each of a, b, c, t has modulus 1, its conjugate is simply its reciprocal, so conjugating the tidy formula for U is just a matter of substituting reciprocals and clearing denominators.

Uˉ=−(E−t)2Q\bar U=-\dfrac{(E-t)^2}{Q}
Detailed analysis

Because ∣a∣=∣b∣=∣c∣=∣t∣=1|a|=|b|=|c|=|t|=1, each variable's conjugate is its reciprocal (aˉ=1/a\bar a=1/a, etc.). Substituting reciprocals into U=−(tS−P)2t2QU=-\dfrac{(tS-P)^2}{t^2Q} and clearing denominators (using tS−P‾=(E−t)/(tP)\overline{tS-P}=(E-t)/(tP) and t2Q‾=Q/(t2P2)\overline{t^2Q}=Q/(t^2P^2)) gives, after simplification, Uˉ=−(E−t)2Q\bar U=-\dfrac{(E-t)^2}{Q}.