MathDB

3

Part of 2001 Federal Competition For Advanced Students, Part 2

Problems(2)

Prove that the three lines have a common point.

Source: Austrian Mathematical Olympiad 2001, Part 2, D1, P3

6/25/2011
A triangle ABCABC is inscribed in a circle with center UU and radius rr. A tangent c′c' to a larger circle K(U,2r)K(U, 2r) is drawn so that C lies between the lines c=ABc = AB and C′C'. Lines a′a' and b′b' are analogously defined. The triangle formed by a′,b′,c′a', b', c' is denoted A′B′C′A'B'C'. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.
geometry unsolvedgeometry
Prove that CF < FD

Source: Austrian Mathematical Olympiad 2001, Part 2, D2, P3

6/25/2011
Let be given a semicircle with the diameter ABAB, and points C,DC,D on it such that AC=CDAC = CD. The tangent at CC intersects the line BDBD at EE. The line AEAE intersects the arc of the semicircle at FF. Prove that CF<FDCF < FD.
geometryangle bisectorgeometry proposed