MathDB

3

Part of 2011 Sharygin Geometry Olympiad

Problems(4)

Two midperpendiculars

Source: 2011 Sharygin Geometry Olympiad #3

5/22/2014
Let ABCABC be a triangle with ∠A=60∘\angle{A} = 60^\circ. The midperpendicular of segment ABAB meets line ACAC at point C1C_1. The midperpendicular of segment ACAC meets line ABAB at point B1B_1. Prove that line B1C1B_1C_1 touches the incircle of triangle ABCABC.
geometrygeometric transformationreflectionangle bisectorgeometry unsolved
3 perpendiculars are concurrent. defined by intersection of // and circumcircle

Source: Sharygin 2011 Final 8.3

12/13/2018
The line passing through vertex AA of triangle ABCABC and parallel to BCBC meets the circumcircle of ABCABC for the second time at point A1A_1. Points B1B_1 and C1C_1 are defined similarly. Prove that the perpendiculars from A1,B1,C1A_1, B_1, C_1 to BC,CA,ABBC, CA, AB respectively concur.
geometryperpendicularconcurrencyconcurrentcircumcircle
isosceles construction given common points of bisectors, medians, altitudes

Source: Sharygin 2011 Final 9.3

12/16/2018
Restore the isosceles triangle ABCABC (AB=ACAB = AC) if the common points I,M,HI, M, H of bisectors, medians and altitudes respectively are given.
geometryisoscelesconstructionIsosceles Triangle
all but one of specific pairs of edges of 2 tetrahedrons are perpendicular

Source: Sharygin 2011 Final 10.3

3/31/2019
Given two tetrahedrons A1A2A3A4A_1A_2A_3A_4 and B1B2B3B4B_1B_2B_3B_4. Consider six pairs of edges AiAjA_iA_j and BkBlB_kB_l, where (i,j,k,li, j, k, l) is a transposition of numbers (1,2,3,41, 2, 3, 4) (for example A1A2A_1A_2 and B3B4B_3B_4). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.
geometrytetrahedronperpendicularsolid geometry