MathDB

2

Part of 2002 Turkey Team Selection Test

Problems(2)

Given cut ratio of two angle bisectors

Source: Turkey TST 2002 - P2

4/6/2013
In a triangle ABCABC, the angle bisector of ABC^\widehat{ABC} meets [AC][AC] at DD, and the angle bisector of BCA^\widehat{BCA} meets [AB][AB] at EE. Let XX be the intersection of the lines BDBD and CECE where ∣BX∣=3∣XD∣|BX|=\sqrt 3|XD| ve ∣XE∣=(3−1)∣XC∣|XE|=(\sqrt 3 - 1)|XC|. Find the angles of triangle ABCABC.
ratiogeometryangle bisectorgeometry proposed
Internally tangent circles

Source: Turkey TST 2002 - P5

4/6/2013
Two circles are internally tangent at a point AA. Let CC be a point on the smaller circle other than AA. The tangent line to the smaller circle at CC meets the bigger circle at DD and EE; and the line ACAC meets the bigger circle at AA and PP. Show that the line PEPE is tangent to the circle through AA, CC, and EE.
geometrygeometric transformationhomothetyreflectionrotationgeometry proposedInversion