MathDB

Problem 3

Part of 1998 Slovenia National Olympiad

Problems(4)

find ratio between sides of rectangle (Slovenia 1998 1st Grade P3)

Source:

4/28/2021
A point EE on side CDCD of a rectangle ABCDABCD is such that △DBE\triangle DBE is isosceles and △ABE\triangle ABE is right-angled. Find the ratio between the side lengths of the rectangle.
geometryrectangle
circumcircle configuration, prove parallel lines

Source: Slovenia 1998 2nd Grade P3

4/28/2021
A point AA is outside a circle K\mathcal K with center OO. Line AOAO intersects the circle at BB and CC, and a tangent through AA touches the circle in DD. Let EE be an arbitrary point on the line BDBD such that DD lies between BB and EE. The circumcircle of the triangle DCEDCE meets line AOAO at CC and FF and line ADAD at DD and GG. Prove that the lines BDBD and FGFG are parallel.
geometrycircumcircle
line through incenters forms a triangle with circumcenter of a vertex

Source: Slovenia 1998 4th Grade P3

4/30/2021
In a right-angled triangle ABCABC with the hypotenuse BCBC, DD is the foot of the altitude from AA. The line through the incenters of the triangles ABDABD and ADCADC intersects the legs of △ABC\triangle ABC at EE and FF. Prove that AA is the circumcenter of triangle DEFDEF.
geometryincentercircumcircleTriangle
circle in rectangle intersecting sides

Source: Slovenia 1998 3rd Grade P3

4/29/2021
A rectangle ABCDABCD with AB>ADAB>AD is given. The circle with center BB and radius ABAB intersects the line CDCD at EE and FF.
(a) Prove that the circumcircle of triangle EBFEBF is tangent to the circle with diameter ADAD. Denote the tangency point by GG. (b) Prove that the points D,G,D,G, and BB are collinear.
geometryrectangle