MathDB

2

Part of 2016 Saudi Arabia BMO TST

Problems(6)

perpendicular wanted, tangents of circle related

Source: 2016 Saudi Arabia BMO TST , level 4, I p2

7/26/2020
Let AA be a point outside the circle ω\omega. Two points B,CB, C lie on ω\omega such that AB,ACAB, AC are tangent to ω\omega. Let DD be any point on ω\omega (DD is neither BB nor CC) and MM the foot of perpendicular from BB to CDCD. The line through DD and the midpoint of BMBM meets ω\omega again at PP. Prove that AP⊥CPAP \perp CP
geometryperpendicularTangents
right angle wanted, 3 circles related

Source: 2016 Saudi Arabia BMO TST , level 4, II p2

7/26/2020
A circle with center OO passes through points AA and CC and intersects the sides ABAB and BCBC of triangle ABCABC at points KK and NN, respectively. The circumcircles of triangles ABCABC and KBNKBN meet at distinct points BB and MM. Prove that ∠OMB=90o\angle OMB = 90^o.
geometrycirclesright angle
collinear wanted, incenter, circle tangent to circumcircle

Source: 2016 Saudi Arabia BMO TST , level 4, III p2

7/26/2020
Let ABCABC be a triangle and II its incenter. The point DD is on segment BCBC and the circle ω\omega is tangent to the circumcirle of triangle ABCABC but is also tangent to DC,DADC, DA at E,FE, F, respectively. Prove that E,FE, F and II are collinear.
geometryincentertangent circlescollinearcircumcircle
perpendicular wanted, incircle and 2 circumcircles related

Source: 2016 Saudi Arabia BMO TST , level 4+, I p2

7/26/2020
Let ABCABC be a triangle with AB≠ACAB \ne AC. The incirle of triangle ABCABC is tangent to BC,CA,ABBC, CA, AB at D,E,FD, E, F, respectively. The perpendicular line from DD to EFEF intersects ABAB at XX. The second intersection point of circumcircles of triangles AEFAEF and ABCABC is TT. Prove that TX⊥TFTX \perp T F
geometrycircumcircleincircleperpendicular
concurrent wanted, excenter and circumcircle related

Source: 2016 Saudi Arabia BMO TST , level 4+, II p2

7/26/2020
Let IaI_a be the excenter of triangle ABCABC with respect to AA. The line AIaAI_a intersects the circumcircle of triangle ABC at TT. Let XX be a point on segment TIaTI_a such that XIa2=XA⋅XTX I_a^2 = XA \cdot X T The perpendicular line from XX to BCBC intersects BCBC at A′A'. Define B′B' and C′C' in the same way. Prove that AA′,BB′AA',BB' and CC′CC' are concurrent.
geometryexcentercircumcircleconcurrentconcurrency
concurrency point is radical center of reflections of excircles wrt midpoints

Source: 2016 Saudi Arabia BMO TST , level 4+, III p2

7/26/2020
Let II be the incenter of an acute triangle ABCABC. Assume that K1K_1 is the point such that AK1⊥BCAK_1 \perp BC and the circle with center K1K_1 of radius K1AK_1A is internally tangent to the incircle of triangle ABCABC at A1A_1. The points B1,C1B_1, C_1 are defined similarly. a) Prove that AA1,BB1,CC1AA_1, BB_1, CC_1 are concurrent at a point PP. b) Let ω1,ω2,ω3\omega_1,\omega_2,\omega_3 be the excircles of triangle ABCABC with respect to A,B,CA, B, C, respectively. The circles γ1,γ2γ3\gamma_1,\gamma_2\gamma_3 are the reflections of ω1,ω2,ω3\omega_1,\omega_2,\omega_3 with respect to the midpoints of BC,CA,ABBC, CA, AB, respectively. Prove that P is the radical center of γ1,γ2,γ3\gamma_1,\gamma_2,\gamma_3.
geometryreflectionincenterRadical centerexcircle