6

Part of 2011 Oral Moscow Geometry Olympiad

Problems(2)

triangle inscribed in triangle, 1 of 2 smallest sides is the inner's triangle

Source: 2011 Oral Moscow Geometry Olympiad grades 8-9 p6

One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.

geometrygeometric inequalityinscribed triangleinscribedside

concurrency wanted, 2 circumcircles and tangents related

Source: 2011 Oral Moscow Geometry Olympiad grades 10-11 p6

AA_1 , BB_1

CC_1

be the altitudes of the non-isosceles acute-angled triangle

ABC

. The circles circumscibred around the triangles

ABC

A_1 B_1 C

intersect again at the point

P , Z

is the intersection point of the tangents to the circumscribed circle of the triangle

ABC

conducted at points

A

B

. Prove that lines

AP , BC

ZC_1

are concurrent.

geometrycircumcirclealtitudesconcurrencyconcurrent