MathDB
4 lines concurrent , starting with semicircle , CD = CH = CZ

Source: 2019 Balkan MO Shortlist G9 - BMO

November 20, 2020
geometrysemicircleconcurrencyconcurrent

Problem Statement

Given semicircle (c)(c) with diameter ABAB and center OO. On the (c)(c) we take point CC such that the tangent at the CC intersects the line ABAB at the point EE. The perpendicular line from CC to ABAB intersects the diameter ABAB at the point DD. On the (c)(c) we get the points H,ZH,Z such that CD=CH=CZCD = CH = CZ. The line HZHZ intersects the lines CO,CD,ABCO,CD,AB at the points S,I,KS, I, K respectively and the parallel line from II to the line ABAB intersects the lines CO,CKCO,CK at the points L,ML,M respectively. We consider the circumcircle (k)(k) of the triangle LMDLMD, which intersects again the lines AB,CKAB, CK at the points P,UP, U respectively. Let (e1),(e2),(e3)(e_1), (e_2), (e_3) be the tangents of the (k)(k) at the points L,M,PL, M, P respectively and R=(e1)∩(e2)R = (e_1) \cap (e_2), X=(e2)∩(e3)X = (e_2) \cap (e_3), T=(e1)∩(e3)T = (e_1) \cap (e_3). Prove that if QQ is the center of (k)(k), then the lines RD,TU,XSRD, TU, XS pass through the same point, which lies in the line IQIQ.
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