MathDB

3

Part of 2015 Iran Geometry Olympiad

Problems(3)

ABC=90^o, BCD=30^o, AB = CD, BC = 2AD, prove BAD=30^o

Source: IGO 2015 Elementary 3

7/22/2018
In the figure below, we know that AB=CDAB = CD and BC=2ADBC = 2AD. Prove that ∠BAD=30o\angle BAD = 30^o. https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png
geometryElementaryangles
line AZ is perpendicular to BC

Source: IGO 2015 Intermediate P3

6/25/2016
In triangle ABCABC ,M,N,KM,N,K are midpoints of sides BC,AC,ABBC,AC,AB,respectively.Construct two semicircles with diameter AB,ACAB,AC outside of triangle ABCABC.MK,MNMK,MN intersect with semicircles in X,YX,Y.The tangents to semicircles at X,YX,Y intersect at point ZZ.Prove that AZ⊥BCAZ \perp BC.(Mehdi E'tesami Fard)
geometrygeometry proposed
IGO - Hard level - Problem 3

Source:

9/3/2015
let H H the orthocenter of the triangle ABC ABC
pass two lines l1 l_1 and l2 l_2 through H H such that l1⊥l2 l_1 \bot l_2
we have l1∩BC=D l_1 \cap BC = D and l1∩AB=Z l_1 \cap AB = Z
also l2∩BC=E l_2 \cap BC = E and l2∩AC=X l_2 \cap AC = X like this picture
pass a line d1 d_1 through D D parallel to AC AC and another line d2 d_2 through E E parallel to AB AB
let d1∩d2=Y d_1 \cap d_2 = Y
prove X X , , Y Y and Z Z are on a same line
IGOgeometry