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Part of 2010 Sharygin Geometry Olympiad

Problems(4)

Does there exist a triangle? (1)

Source:

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

geometrygeometry unsolved

circumcircle tangent to angle bisector

Source: Sharygin 2010 Final 8.1

For a nonisosceles triangle

ABC

, consider the altitude from vertex

A

and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex

A

geometrycircumcircleangle bisectortangent

angle chasing, related to angle between altitude and angle bisector

Source: Sharygin 2010 Final 9.1

For each vertex of triangle

ABC

, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices

A

B

were equal. Furthermore the angle in vertex

C

is greater than two remaining angles. Find angle

C

of the triangle.

geometryangle bisectorAngle Chasinganglesaltitude

find OJ where J is symmetric of a right angle wrt I, related to R,r

Source: Sharygin 2010 Final 10.1

O, I

be the circumcenter and the incenter of a right-angled triangle,

R, r

be the radii of respective circles,

J

be the reflection of the vertex of the right angle in

I

OJ

geometrycircumradiusinradiusMetric Relation