MathDB

6

Part of 2014 Oral Moscow Geometry Olympiad

Problems(2)

angle chasing candidate, inside a right and isosceles triangle

Source: 2014 Oral Moscow Geometry Olympiad grades 8-9 p6

8/9/2019
Inside an isosceles right triangle ABCABC with hypotenuse ABAB a point MM is taken such that the angle ∠MAB\angle MAB is 15o15 ^o larger than the angle ∠MAC\angle MAC , and the angle ∠MCB\angle MCB is 15o15^o larger than the angle ∠MBC\angle MBC. Find the angle ∠BMC\angle BMC .
geometryanglesAngle Chasingright triangleisosceles
2 lines connecting incenters with excenters concurrent with angle bisector

Source: 2014 Oral Moscow Geometry Olympiad grades 10-11 p6

8/9/2019
A convex quadrangle ABCDABCD is given. Let II and JJ be the circles of circles inscribed in the triangles ABCABC and ADCADC, respectively, and IaI_a and JaJ_a are the centers of the excircles circles of triangles ABCABC and ADCADC, respectively (inscribed in the angles BACBAC and DACDAC, respectively). Prove that the intersection point KK of the lines IJaIJ_a and JIaJI_a lies on the bisector of the angle BCDBCD.
geometryincenterexcenterconcurrencyconcurrentangle bisector