MathDB

2

Part of 2003 Hungary-Israel Binational

Problems(2)

$CC_1$ bisects $\widehat{QC_1P}$ .

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

3/30/2007
Let ABCABC be an acute-angled triangle. The tangents to its circumcircle at A,B,CA, B, C form a triangle PQRPQR with C∈PQC \in PQ and B∈PRB \in PR. Let C1C_{1} be the foot of the altitude from CC in ΔABC\Delta ABC . Prove that CC1CC_{1} bisects QC1P^\widehat{QC_{1}P} .
geometrycircumcircletrigonometrygeometry unsolved
one of the lines $AA_1 , BB_1, CC_1$ is a median of ABC

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

3/30/2007
Let MM be a point inside a triangle ABCABC . The lines AM,BM,CMAM , BM , CM intersect BC,CA,ABBC, CA, AB at A1,B1,C1A_{1}, B_{1}, C_{1}, respectively. Assume that SMAC1+SMBA1+SMCB1=SMA1C+SMB1A+SMC1BS_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B} . Prove that one of the lines AA1,BB1,CC1AA_{1}, BB_{1}, CC_{1} is a median of the triangle ABC.ABC.
geometry proposedgeometry