## Search found 48 matches

Thu Feb 16, 2017 9:45 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 52527

### Re: Geometry Marathon : Season 3

Prooblem 28 :

Let $ABC$ be a triangle with altitudes $AD,BE,CF$. $X,Y,Z$ are midpoints of $BC,CA,AB$. Consider circles $(X,XD),(Y,YE),(Z,ZF)$. Prove that radical center of $(X),(Y),(Z)$ is Nagel point of triangle $DEF$.
Thu Feb 16, 2017 1:28 am
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 52527

### Re: Geometry Marathon : Season 3

Solution of problem 27 : Let $I_a,I_b,I_C$ be the excenters of $ABC$ opposite to $A,B,C$ respectively . $O$ be the center of $(I_aI_bI_C)$. $I_aO \cap BC=A_1$.Let $I_aO \cap (I_aI_bI_C)=A_2 ,I_a$.let $A_2I \cap (I_aI_bI_C)=A_2,A_3$ and $I_aA_3\cap I_bI_c=A_4,$Let points $A_5,A_6 \in I_bI_c$, such t...
Tue Feb 14, 2017 9:08 pm
Topic: BdMO 2017 National Round Secondary 9
Replies: 2
Views: 1636

Let $QO$ meet $(DQC)$ again at $O'$, then $\angle DQX = \angle CQY \Rightarrow DO'=O'C$ implying $O=O'$. Then $FD .FC=FQ.FO \Rightarrow Q$ is the invertion of $F$ wrt $(ABCD)$ ,by brocard's theorm $EQ$ is the polar of $F$ wrt $(ABCD) \Rightarrow -1=(EF,EP;ED,EC)=(PF,PE;PD,PC)$. $\angle EPD = \angle... Mon Feb 13, 2017 1:36 am Forum: Geometry Topic: Geometry Marathon : Season 3 Replies: 110 Views: 52527 ### Re: Geometry Marathon : Season 3 Solution of problem 22: Lemma : Let$ \omega_1$&$ \omega_2$be two circles with center$O_1 $&$O_2$respectively.$ \omega_1 \cap \omega_2 = A ,B$.Let the tangents to$ \omega_1$at$A , B$meet at$E$.Let$C,D \in \omega_2$such that$ACBD$is harmonic .$CE \cap \omega_2 = C,F $and$FO_2 \cap...
Mon Feb 06, 2017 10:18 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 52527

### Re: Geometry Marathon : Season 3

Problem 22:

Prove that the circumcenter of $\triangle ABC$ and the centroid of the anti pedal $\triangle A'B'C'$ of the symmedian point of $\triangle ABC$ coincide.
Mon Feb 06, 2017 10:01 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 110
Views: 52527