MathDB

7

Part of 2015 Sharygin Geometry Olympiad

Problems(3)

2 tangential quadrilaterals from vertices of a quadrilateral & a point on 1 side

Source: Sharygin Geometry Olympiad 2015 Final 8.7

8/1/2018
Point MM on side ABAB of quadrilateral ABCDABCD is such that quadrilaterals AMCDAMCD and BMDCBMDC are circumscribed around circles centered at O1O_1 and O2O_2 respectively. Line O1O2O_1O_2 cuts an isosceles triangle with vertex M from angle CMDCMD. Prove that ABCDABCD is a cyclic quadrilateral.
(M. Kungozhin)
geometrycyclic quadrilateraltangential quadrilateral
segment of orthocenter - centroid, bisects segment of feet of altitudes

Source: Sharygin Geometry Olympiad 2015 Final 9.7

8/1/2018
Let ABCABC be an acute-angled, nonisosceles triangle. Altitudes AA′AA' and BB′BB' meet at point HH, and the medians of triangle AHBAHB meet at point MM. Line CMCM bisects segment A′B′A'B'. Find angle CC.
(D. Krekov)
geometryangleorthocenterCentroid
Sphere geometry

Source: Sharygin geometry olympiad 2015, grade 10, Final Round, Problem 7

7/17/2018
Let SABCDSABCD be an inscribed pyramid, and AA1AA_1, BB1BB_1, CC1CC_1, DD1DD_1 be the perpendiculars from AA, BB, CC, DD to lines SCSC, SDSD, SASA, SBSB respectively. Points SS, A1A_1, B1B_1, C1C_1, D1D_1 are distinct and lie on a sphere. Prove that points A1A_1, B1B_1, C1C_1 and D1D_1 are coplanar.
geometry3D geometrysphere