## Search found 48 matches

Sat Apr 22, 2017 7:04 pm
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 9547

### 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: 1411

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 .$\ang... Mon Feb 27, 2017 9:13 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Problem 38 : In$\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$. Mon Feb 27, 2017 7:57 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Solution of problem 36: Let$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$is the reflec... Sun Feb 26, 2017 3:37 pm Forum: Geometry Topic: IGO 2016 Elementary/2 Replies: 5 Views: 2115 ### Re: IGO 2016 Elementary/2 As,$ \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}$Sat Feb 25, 2017 6:02 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Problem 34: Let$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$. Thu Feb 23, 2017 10:22 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Solution of problem 33: Let$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$at ... Tue Feb 21, 2017 4:10 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Problem 32 : In triangle$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$similar... Tue Feb 21, 2017 4:00 pm Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52510 ### Re: Geometry Marathon : Season 3 Solution of problem 31: Let$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...
Sat Feb 18, 2017 1:51 am
Forum: Geometry
Topic: IMO Shortlist 2010 G7
Replies: 1
Views: 1382

### Re: IMO Shortlist 2010 G7

Lemma : Let two circular arcs $\alpha$ & $\beta$ connect pionts $A,B$. If two circle $\varpi_1$ & $\varpi_2$ are tangent to $\alpha$ & $\beta$ ,then their external center of simillitude lies in $AB$. Proof : Let an external common tangent $l$ of $\alpha$ & $\beta$ meet $AB$ at piont $M$.Then t...