MathDB

5

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

Prove that the points L,E,F are collinear (5)

Source:

10/28/2010
A point EE lies on the altitude BDBD of triangle ABCABC, and ∠AEC=90∘.\angle AEC=90^\circ. Points O1O_1 and O2O_2 are the circumcenters of triangles AEBAEB and CEBCEB; points F,LF, L are the midpoints of the segments ACAC and O1O2.O_1O_2. Prove that the points L,E,FL,E,F are collinear.
geometrycircumcirclegeometric transformationdilationgeometry proposed
altitude, bisector and median of HLM, constructed by altitude, median, bisector

Source: Sharygin 2010 Final 8.5

10/2/2018
Let AHAH, BLBL and CMCM be an altitude, a bisectrix and a median in triangle ABCABC. It is known that lines AHAH and BLBL are an altitude and a bisectrix of triangle HLMHLM. Prove that line CMCM is a median of this triangle.
geometryaltitudemedianangle bisector
touchpoints of circumcircles defined by altitude of right angle

Source: Sharygin 2010 Final 10.5

11/25/2018
Let BHBH be an altitude of a right-angled triangle ABCABC (∠B=90o\angle B = 90^o). The incircle of triangle ABHABH touches AB,AHAB,AH in points H1,B1H_1, B_1, the incircle of triangle CBHCBH touches CB,CHCB,CH in points H2,B2H_2, B_2, point OO is the circumcenter of triangle H1BH2H_1BH_2. Prove that OB1=OB2OB_1 = OB_2.
geometryequal segmentsright trianglealtitudecircumcircle
angle chasing with touchpoints from incircle and excirles

Source: Sharygin 2010 Final 9.5

10/6/2018
The incircle of a right-angled triangle ABCABC (∠ABC=90o\angle ABC =90^o) touches AB,BC,ACAB, BC, AC in points C1,A1,B1C_1, A_1, B_1, respectively. One of the excircles touches the side BCBC in point A2A_2. Point A0A_0 is the circumcenter or triangle A1A2B1A_1A_2B_1, point C0C_0 is defined similarly. Find angle A0BC0A_0BC_0.
geometryincircleAngle Chasingexcircle