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AX and HT meet on circumcircle

Source: 2022 Israel TST test 10 P3

July 18, 2022
geometryTSTIsraelcircumcircleincenter

Problem Statement

Scalene triangle ABCABC has incenter II and circumcircle Ω\Omega with center OO. HH is the orthocenter of triangle BICBIC, and TT is a point on Ω\Omega for which ∠ATI=90∘\angle ATI=90^\circ. Circle (AIO)(AIO) intersects line IHIH again at XX. Show that the lines AX,HTAX, HT intersect on Ω\Omega.
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