MathDB

6

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Covering bisected triangles with semi-circumdisc

Source: Sharygin Finals 2017, Problem 8.6

8/4/2017
A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.
geometrycoveringmediancircumcircle
Ratio of areas erected on right triangle

Source: Sharygin Finals 2017, Problem 9.6

8/3/2017
Let ABCABC be a right-angled triangle (∠C=90∘\angle C = 90^\circ) and DD be the midpoint of an altitude from C. The reflections of the line ABAB about ADAD and BDBD, respectively, meet at point FF. Find the ratio SABF:SABCS_{ABF}:S_{ABC}. Note: SαS_{\alpha} means the area of α\alpha.
ratiogeometryareas
Inspheres and sum of angles

Source: Sharygin 2017 Day 2 Problem 10.6 Grade 10

8/2/2017
10.6 Let the insphere of a pyramid SABCSABC touch the faces SAB,SBC,SCASAB, SBC, SCA at D,E,FD, E, F respectively. Find all the possible values of the sum of the angles SDA,SEB,SFCSDA, SEB, SFC.
geometry3D geometry