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Nice I configuration with orthocenters

Source: Turkey TST 2022 P8 Day 3

March 13, 2022
geometrygeometry proposedorthocenterincircle

Problem Statement

ABCABC triangle with ∣AB∣<∣BC∣<∣CA∣|AB|<|BC|<|CA| has the incenter II. The orthocenters of triangles IBC,IACIBC, IAC and IABIAB are HA,HAH_A, H_A and HAH_A. HBHCH_BH_C intersect BCBC at KAK_A and perpendicular line from II to HBHBH_BH_B intersect BCBC at LAL_A. KB,LB,KC,LCK_B, L_B, K_C, L_C are defined similarly. Prove that ∣KALA∣=∣KBLB∣+∣KCLC∣|K_AL_A|=|K_BL_B|+|K_CL_C|
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