MathDB

6

Part of 2012 Oral Moscow Geometry Olympiad

Problems(2)

triangle construction, given orthocenter, cuts of median, angle-bisector

Source: 2012 Oral Moscow Geometry Olympiad grades 8-9 p6

9/8/2019
Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)
geometryconstructionorthocentermedianangle bisector
tangent circumcircles wanted

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p6

9/25/2019
Tangents drawn to the circumscribed circle of an acute-angled triangle ABCABC at points AA and CC, intersect at point ZZ. Let AA1,CC1AA_1, CC_1 be altitudes. Line A1C1A_1C_1 intersects ZA,ZCZA, ZC at points XX and YY, respectively. Prove that the circumscribed circles of the triangles ABCABC and XYZXYZ are tangent.
geometrycircumcircletangent circles