MathDB

2

Part of 2014 CentroAmerican

Problems(2)

Trapezoid and circles

Source: Centroamerican Olympiad 2014, problem 2

6/10/2014
Let ABCDABCD be a trapezoid with bases ABAB and CDCD, inscribed in a circle of center OO. Let PP be the intersection of the lines BCBC and ADAD. A circle through OO and PP intersects the segments BCBC and ADAD at interior points FF and GG, respectively. Show that BF=DGBF=DG.
geometrytrapezoidgeometry proposed
Three equilateral triangles

Source: Centroamerican Olympiad 2014, problem 5

6/11/2014
Points AA, BB, CC and DD are chosen on a line in that order, with ABAB and CDCD greater than BCBC. Equilateral triangles APBAPB, BCQBCQ and CDRCDR are constructed so that PP, QQ and RR are on the same side with respect to ADAD. If ∠PQR=120∘\angle PQR=120^\circ, show that
1AB+1CD=1BC.\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.
geometryincentergeometric transformationreflectiontrigonometrycircumcircleangle bisector