## Search found 138 matches

Sun Nov 26, 2017 11:16 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 11316

### Re: Geometry Marathon : Season 3

$\text{Problem 50:}$ Let $\vartriangle ABC$ be a triangle, $O$ be the circumcenter of $\vartriangle ABC$, $N$ be the center of nine point circle of $\vartriangle ABC$ and $X$ be the midpoint of the line segment $ON$. Let $A',B',C'$ be the midpoints of the line segments $BC,CA,AB$, respectively. Let ...
Sun Nov 26, 2017 10:48 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 11316

$\text{Solution to Problem 49:}$ Let $BH \cap AC=E, CH \cap AB=F, AP \cap BC=X$. Now, by miquel's theorem on circle $BCEF, M,H,P$ are collinear. Also by radical axis theorem on circle $ABC, APFHE, BCEF$, we get $X,F,E$ are collinear. Now, we get that $H$ is the orthocentre of triangle $AXM$ so $XH \... Sat Nov 25, 2017 11:59 am Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 11316 ### Re: Geometry Marathon : Season 3$\text{Problem 48:}$Prove that for a scalene triangle$ABC$, one can never find two pairs of isogonal conjugates${P,P'}$and${Q,Q'}$such that$P,P',Q,Q'$are all collinear and given that they're not self isogonal conjugates. Sat Nov 25, 2017 11:56 am Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 11316 ### Re: Geometry Marathon : Season 3$\text{Solution to Problem 47:}$Let$AC \cap BD=X$,$AC \cap EF=Y$and$BD \cap EF=Z$. Now$P,M,N$are collinear by newton line. Then,$XM.XZ=XB.XD=XA.XC=XN.XY \Rightarrow MNYZ$is cyclic and$PE^2=PY.PZ=PN.PM$. Sat Oct 21, 2017 4:20 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 11316 ### Re: Geometry Marathon : Season 3$\text{Problem 45}$Let$ABC$be a triangle with orthocentre$H$and circumcircle$\omega$centered at$O$. Let$M_a,M_b,M_c$be the midpoints of$BC,CA,AB$. Lines$AM_a,BM_b,CM_c$meet$\omega$again at$P_a,P_b,P_c$. Rays$M_aH,M_bH,M_cH$intersect$\omega$at$Q_a,Q_b,Q_c$. Prove that$P_aQ_a,P_b...
Sat Oct 21, 2017 3:58 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 11316