MathDB

3.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(2)

Concyclic points and a bisector

Source: Saint Petersburg MO 2020 Grade 11 Problem 3

5/7/2020
BB1BB_1 is the angle bisector of △ABC\triangle ABC, and II is its incenter. The perpendicular bisector of segment ACAC intersects the circumcircle of △AIC\triangle AIC at DD and EE. Point FF is on the segment B1CB_1C such that AB1=CFAB_1=CF.Prove that the four points B,D,EB, D, E and FF are concyclic.
geometryangle bisectorincenterperpendicular bisectorcircumcircle
Geometric inequality

Source: Saint Petersburg MO 2020 Grade 9 Problem 3 & Grade 10 Problem 3

5/7/2020
On the side ADAD of the convex quadrilateral ABCDABCD with an acute angle at BB, a point EE is marked. It is known that ∠CAD=∠ADC=∠ABE=∠DBE\angle CAD = \angle ADC=\angle ABE =\angle DBE. (Grade 9 version) Prove that BE+CE<ADBE+CE<AD. (Grade 10 version) Prove that △BCE\triangle BCE is isosceles.(Here the condition that ∠B\angle B is acute is not necessary.)
geometric inequalityinequalities