MathDB

12

Part of 2018 Hanoi Open Mathematics Competitions

Problems(2)

right angle inside a rectangle wanted (HOMC 2018 Ind. p12)

Source:

1/31/2020
Let ABCD be a rectangle with 45o<∠ADB<60o45^o < \angle ADB < 60^o. The diagonals ACAC andBD BD intersect at OO. A line passing through OO and perpendicular to BDBD meets ADAD and CDCD at MM and NN respectively. Let KK be a point on side BCBC such that MK∥ACMK \parallel AC. Show that ∠MKN=90o\angle MKN = 90^o. https://cdn.artofproblemsolving.com/attachments/4/1/1d37b96cebaea3409ade7ce6711ac2d3fc2ef9.png
geometryrectangle
AC is tangent to circumcircle of DEF (HOMC 2018 ind. sen12)

Source:

2/2/2020
Let ABCABC be an acute triangle with AB<ACAB < AC, and let BEBE and CFCF be the altitudes. Let the median AMAM intersect BEBE at point PP, and let line CPCP intersect ABAB at point DD (see Figure 2). Prove that DE∥BCDE \parallel BC, and ACAC is tangent to the circumcircle of △DEF\vartriangle DEF. https://cdn.artofproblemsolving.com/attachments/f/7/bbad9f6019a77c6aa46c3a821857f06233cb93.png
geometrycircumcircletangentparallelaltitudes