MathDB

g10

Part of Indonesia MO Shortlist - geometry

Problems(4)

BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector

Source: Indonesia INAMO Shortlist 2008 G10

8/25/2021
Given a triangle ABCABC with AB=ACAB = AC, angle ∠A=100o\angle A = 100^o and BDBD bisector of angle ∠B\angle B. Prove that BC=BD+DA.BC = BD + DA.
geometryequal segmentsisosceles
incenter of ABC is orthocenter of triangles of excenters

Source: Indonesia INAMO Shortlist 2009 G10 https://artofproblemsolving.com/community/c1101409_

12/10/2021
Given a triangle ABCABC with incenter II . It is known that EAE_A is center of the ex-circle tangent to BCBC. Likewise, EBE_B and ECE_C are the centers of the ex-circles tangent to ACAC and ABAB, respectively. Prove that II is the orthocenter of the triangle EAEBECE_AE_BE_C.
geometryincenterexcirclesexcenters
incenter wanted, 2 intersecting circles, one has center on the other circle

Source: Indonesia INAMO Shortlist 2010 G10

8/27/2021
Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points CC and DD. The line through DD intersects the two circles again at AA and B B. Let HH be the midpoint of the arc ACAC that does not contain DD and the segment HDHD intersects circle that does not contain HH at point EE. Show that EE is the center of the incircle of the triangle ACDACD.
geometryincentercircles
OD _|_ MN iff P,Q,M,N concyclic, 2 intersecting circles

Source: Indonesia INAMO Shortlist 2017 G10 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
It is known that circle Γ1(O1)\Gamma_1(O_1) has center at O1O_1, circle Γ2(O2)\Gamma_2(O_2) has center at O2O_2, and both intersect at points CC and DD. It is also known that points PP and QQ lie on circles Γ1(O1)\Gamma_1(O_1) and Γ2(O2)\Gamma_2(O_2), respectively. ). A line ℓ\ell passes through point DD and intersects Γ1(O1)\Gamma_1(O_1) and Γ2(O2)\Gamma_2(O_2) at points AA and BB, respectively. The lines PDPD and ACAC meet at point MM, and the lines QDQD and BCBC meet at point NN. Let OO be center outer circle of triangle ABCABC. Prove that ODOD is perpendicular to MNMN if and only if a circle can be found which passes through the points P,Q,MP, Q, M and NN.
perpendicularConcyclicgeometry