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Part of 2014 Vietnam Team Selection Test

Problems(1)

Viet Nam TST 2014 day 2 problem 1

Source:

3/28/2014
a. Let ABCABC be a triangle with altitude ADAD and PP a variable point on ADAD. Lines PBPB and ACAC intersect each other at EE, lines PCPC and ABAB intersect each other at F.F. Suppose AEDFAEDF is a quadrilateral inscribed . Prove that PAPD=(tan⁡B+tan⁡C)cot⁡A2.\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}. b. Let ABCABC be a triangle with orthocentre HH and PP a variable point on AHAH. The line through CC perpendicular to ACAC meets BPBP at MM, The line through BB perpendicular to ABAB meets CPCP at N.N. KK is the projection of AAon MNMN. Prove that ∠BKC+∠MAN\angle BKC+\angle MAN is invariant .
trigonometryinvariantgeometry unsolvedgeometry