Search found 49 matches
- Sat Apr 29, 2017 10:09 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
As problem 38 remained unsolved for many days , i will give another problem. (I will try to find a solution for this later when I get enough time . ) Problem 39 : Let $ABC$ be a triangle with circumcenter $O$.Let $BC$ be the smallest side. $(BOC)$ cuts $CA, AB$ at $E$ and $F$. $BE$ cuts $CF$ at $M$....
- Sat Apr 22, 2017 7:04 pm
- Forum: Geometry
- Topic: USA(J)MO 2017 #3
- Replies: 6
- Views: 13366
Re: USA(J)MO 2017 #3
For those who loves synthetic geometry Throughout the proof signed area will be used. Lemma : Let $ABC$ be an equilateral triangle, and point $P$ on its circumcircle. Let $PB$ and $AC$ intersect at $E$, and $PC$ and $AB$ intersect at $F$.Then $ {[EPF]}={[ABPC]}$ Proof: Let the tangent to $(ABC)$ at ...
- Sun Apr 02, 2017 11:34 pm
- Forum: Secondary Level
- Topic: Find the angle
- Replies: 2
- Views: 3045
Re: Find the angle
Let $O$ be the circumcenter of $\triangle ABC$, Then $\angle ABC =180^{\circ}-30^{\circ} -70^{\circ}=80^{\circ} ,\angle OAC =90^{\circ}-\angle ABC=10^{\circ}.$ So $O \in AM$ .$\angle OBC=\angle ABC -\angle ABO =80^{\circ}-20^{\circ}=60^{\circ} $ .As $OB=OC$ so , $\triangle OBC$ is equilateral .$\ang...
- Mon Feb 27, 2017 9:13 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Problem 38 :
In $\triangle ABC$ let the angle bisector of $\angle BAC$ meet $BC$ at $A_o$. Define $B_o,C_o$ similarly.Prove that
the circumcircle of $\triangle A_oB_oC_o$ goes though the Feuerbach point of $\triangle ABC$.
In $\triangle ABC$ let the angle bisector of $\angle BAC$ meet $BC$ at $A_o$. Define $B_o,C_o$ similarly.Prove that
the circumcircle of $\triangle A_oB_oC_o$ goes though the Feuerbach point of $\triangle ABC$.
- Mon Feb 27, 2017 7:57 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Solution of problem 36: Let $S$ be the midpoint of arc $AC$ (containing $B$) & $Q$ be the midpoint of arc $AC$ (not containing $B$).$R$ be the reflection of point $P$ wrt $XY$.Now $M$ be the midpoint of $AC$ & $K$ be the orthocenter of $\triangle SAC $.$N$ be the midpoint of $BH$. $K$ is the reflec...
- Sun Feb 26, 2017 3:37 pm
- Forum: Geometry
- Topic: IGO 2016 Elementary/2
- Replies: 5
- Views: 4614
Re: IGO 2016 Elementary/2
As, $ \angle YXC = \angle CPY + \angle PCA$. So ,$ \angle YXC = \angle XYC \Rightarrow \angle CPY + \angle PCA = \angle XYC \Rightarrow \widehat{AP} + \widehat{CY} = \widehat{PC}$
- Sat Feb 25, 2017 6:02 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Problem 34:
Let $O$ & $I$ denote the circumcenter & incenter of $\triangle ABC$ respectively.Prove that The reflections of the
$OI$ line in the sides of the intouch triangle of $\triangle ABC$ concur at the Feuerbach point of $\triangle ABC$.
Let $O$ & $I$ denote the circumcenter & incenter of $\triangle ABC$ respectively.Prove that The reflections of the
$OI$ line in the sides of the intouch triangle of $\triangle ABC$ concur at the Feuerbach point of $\triangle ABC$.
- Thu Feb 23, 2017 10:22 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Solution of problem 33: Let $I_a$ denote the excenter opposite to $A$ .The $A$- excircle touches $BC$ at $P$ .Let $A_0D \cap OI = J$ & $AI \cap BC=K$.Let the perpendiculer from $O$ to $BC$ meet $AI$ & $A_0D$ at $M,N$ respectively. Lemma : $A_0 ,D ,I_a$ are collinear. Proof :Let $I_aD$ meet $AH$ at ...
- Tue Feb 21, 2017 4:10 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Problem 32 : In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear incircles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similar...
- Tue Feb 21, 2017 4:00 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 191493
Re: Geometry Marathon : Season 3
Solution of problem 31: Let $D,E,F $ be the midpoints of the arc $BC$ (not containg $A$) ,arc $CA$ (not containg $B$),arc $AB$ (not containg $C$) respectively. Let $H_a,H_b,H_c$ be the orthocenters of $\triangle IBC ,\triangle ICA ,\triangle IAB$ respectively .$M ,N ,P$ be the midpoints of $BC,CA,A...