MathDB

4

Part of 2016 Iranian Geometry Olympiad

Problems(3)

angles in right triangle, with perpendicular bisectors

Source: IGO Elementary 2016 4

7/22/2018
In a right-angled triangle ABCABC (∠A=90o\angle A = 90^o), the perpendicular bisector of BCBC intersects the line ACAC in KK and the perpendicular bisector of BKBK intersects the line ABAB in LL. If the line CLCL be the internal bisector of angle CC, find all possible values for angles BB and CC.
by Mahdi Etesami Fard
geometryright triangleperpendicular bisector
Similarity through arc midpoint in right triangle

Source: Iranian Geometry Olympiad 2016 Medium 4

5/26/2017
Let ω\omega be the circumcircle of right-angled triangle ABCABC (∠A=90∘\angle A = 90^{\circ}). The tangent to ω\omega at point AA intersects the line BCBC at point PP. Suppose that MM is the midpoint of the minor arc ABAB, and PMPM intersects ω\omega for the second time in QQ. The tangent to ω\omega at point QQ intersects ACAC at KK. Prove that ∠PKC=90∘\angle PKC = 90^{\circ}.
Proposed by Davood Vakili
geometrycircumcircle
Iranian Geometry Olympiad (4)

Source: Advanced level,P4

9/13/2016
In a convex quadrilateral ABCDABCD, the lines ABAB and CDCD meet at point EE and the lines ADAD and BCBC meet at point FF. Let PP be the intersection point of diagonals ACAC and BDBD. Suppose that ω1\omega_1 is a circle passing through DD and tangent to ACAC at PP. Also suppose that ω2\omega_2 is a circle passing through CC and tangent to BDBD at PP. Let XX be the intersection point of ω1\omega_1 and ADAD, and YY be the intersection point of ω2\omega_2 and BCBC. Suppose that the circles ω1\omega_1 and ω2\omega_2 intersect each other in QQ for the second time. Prove that the perpendicular from PP to the line EFEF passes through the circumcenter of triangle XQYXQY . Proposed by Iman Maghsoudi
geometry