## Search found 138 matches

Tue Feb 21, 2017 2:52 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63421

### Re: Geometry Marathon : Season 3

$\text{Solution to Problem 30: }$ Easy angle chase gives that $K$ is the midpoint of $\text{arc BC}$ not containing $A$.Now, Let $M$ be the midpoint of $BC$ and $L$ be the midpoint of $\text{arc BC}$ containing $A$. We get, $S,E,L$ are collinear and $ALME$ is cyclic. So radical axis theorem gives us...
Mon Feb 20, 2017 4:16 pm
Forum: Geometry
Replies: 2
Views: 1721

Let $ABCD$ be a cyclic quadrilateral. Let $H_A, H_B, H_C, H_D$ denote the orthocenters of triangles $BCD, CDA, DAB$ and $ABC$ respectively. Prove that $AH_A, BH_B, CH_C$ and $DH_D$ concur.
Fri Feb 17, 2017 1:35 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63421

$\text{Solution of Problem 28 :}$ Let, $T$ be the nine-point circle of $\bigtriangleup ABC$. Let $l_y$ be the radical axis of $T,(Y)$ and $l_z$ be the radical axis of $T,(Z)$.And let $A_1$=$l_y \cap l_z$. And the nagel point and centroid of $\bigtriangleup DEF$ be $M$ and $P$ respectively. $\text{L... Tue Feb 14, 2017 1:28 am Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 146 Views: 63421 ### Re: Geometry Marathon : Season 3$\text{Problem 27:}$Let$ABC$be a scalene triangle, let$I$be its incentre , and let$A_1,B_1$and$C_1$be the points of contact of the excircles with the sides$BC,CA$and$AB$respectively. Prove that the circumcircles of the triangles$AIA_1,BIB_1$and$CIC_1$have a common point different fr... Mon Feb 13, 2017 11:24 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 146 Views: 63421 ### Re: Geometry Marathon : Season 3$\text{**The Problem 24 has been posted before here as Problem no 4.}\text{So I'll just give the solution of Problem 25**}\text{Solution of Problem 25 :}$Let,$A_1$be a point such that$\bigtriangleup A_1BC$is equilateral where$A_1$lies on the side of$BC$oppostie to$A$. We define$B_1,...
Mon Feb 13, 2017 1:59 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63421

### Re: Geometry Marathon : Season 3

$\text{Problem 23:}$ Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ agai...
Sat Feb 04, 2017 1:02 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 63421