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2018 VNTST Problem 6

Source: 2018 Vietnam Team Selection Test

March 31, 2018
geometryincircleexcircle

Problem Statement

Triangle ABCABC circumscribed (O)(O) has AA-excircle (J)(J) that touches AB, BC, ACAB,\ BC,\ AC at F, D, EF,\ D,\ E, resp.
a. LL is the midpoint of BCBC. Circle with diameter LJLJ cuts DE, DFDE,\ DF at K, HK,\ H. Prove that (BDK), (CDH)(BDK),\ (CDH) has an intersecting point on (J)(J). b. Let EF∩BC={G}EF\cap BC =\{G\} and GJGJ cuts AB, ACAB,\ AC at M, NM,\ N, resp. P∈JBP\in JB and Q∈JCQ\in JC such that ∠PAB=∠QAC=90∘.\angle PAB=\angle QAC=90{}^\circ . PM∩QN={T}PM\cap QN=\{T\} and SS is the midpoint of the larger BCBC-arc of (O)(O). (I)(I) is the incircle of ABCABC. Prove that SI∩AT∈(O)SI\cap AT\in (O).
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