MathLabs

Problem 6

Let AH1,BH2,CH3 AH_1,BH_2,CH_3 be the altitudes of an acute triangle ABC ABC . The incircle of ABC ABC touches BC,CA,AB BC,CA,AB at T1,T2,T3 T_1,T_2,T_3, respectively. Reflect the lines H1H2,H2H3,H3H1 H_1H_2,H_2H_3,H_3H_1 in the lines T1T2,T2T3,T3T1 T_1T_2,T_2T_3,T_3T_1, respectively. Prove that the three reflected lines form a triangle whose vertices lie on the incircle.
Step 3 of 5: Compute intersections
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V1=abc,V2=bca,V3=cab.V_1=\frac{ab}{c},\quad V_2=\frac{bc}{a},\quad V_3=\frac{ca}{b}.
Detailed analysis

Substitute the tangent and altitude equations into the reflection map; cyclic permutation gives the three pairwise intersections.