MathDB

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Part of 2007 Sharygin Geometry Olympiad

Problems(4)

triangle split into isosceles and equilateral triangles

Source: Sharygin 2007 Correspodence p1

4/30/2019
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
geometryEquilateralEquilateral TriangleisoscelesIsosceles Triangleangles
inradius of a tangential ABCD smaller than sum of inradii of ABC and ACD

Source: Sharygin Final 2007 9.1

4/30/2019
Given a circumscribed quadrilateral ABCDABCD. Prove that its inradius is smaller than the sum of the inradii of triangles ABCABC and ACDACD.
geometryinradiusradiigeometric inequality
determine on which side is the steering wheel disposed in the car

Source: Sharygin Final 2007 8.1

4/30/2019
Determine on which side is the steering wheel disposed in the car depicted in the figure. https://4.bp.blogspot.com/-s2rjZw-d4UY/XMg5BXCE9SI/AAAAAAAAKHc/WOpvqjWw7lAciDEiNj_TX7io6sfItSPnQCK4BGAYYCw/s320/Sharygin%2Bfinal%2B2007%2B8.1.png
geometry
triangle construction given intersections of 2 alt. and bisector w. circumcircle

Source: Sharygin 2007 Final 10.1

4/30/2019
In an acute triangle ABCABC, altitudes at vertices AA and BB and bisector line at angle CC intersect the circumcircle again at points A1,B1A_1, B_1 and C0C_0. Using the straightedge and compass, reconstruct the triangle by points A1,B1A_1, B_1 and C0C_0.
geometryconstructionaltitudesangle bisectorcircumcircle