Found problems: 18
2018 China Western Mathematical Olympiad, 4
In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.
2016 Oral Moscow Geometry Olympiad, 5
Points $I_A, I_B, I_C$ are the centers of the excircles of $ABC$ related to sides $BC, AC$ and $AB$ respectively. Perpendicular from $I_A$ to $AC$ intersects the perpendicular from $I_B$ to $B_C$ at point $X_C$. The points $X_A$ and $X_B$. Prove that the lines $I_AX_A, I_BX_B$ and $I_CX_C$ intersect at the same point.
Russian TST 2021, P2
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
1982 Brazil National Olympiad, 1
The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$ are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.
2012 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$
2021 Taiwan TST Round 3, G
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
1994 Korea National Olympiad, Problem 3
In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that:
(i) $R'= 2R $
(ii) $IO' = 2IO$
2021 SAFEST Olympiad, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3
Denote in the triangle $ABC$ by $T_A,T_B,T_C$ the touch points of the exscribed circles of $\vartriangle ABC$, tangent to sides $BC, AC$ and $AB$ respectively. Let $O$ be the center of the circumcircle of $\vartriangle ABC$, and $I$ is the center of it's inscribed circle. It is known that $OI\parallel AC$. Prove that $\angle T_A T_B T_C= 90^o - \frac12 \angle ABC$.
(Anton Trygub)
2003 IMO Shortlist, 3
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
Indonesia MO Shortlist - geometry, g10
Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.
2004 Germany Team Selection Test, 2
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
2021 Saudi Arabia IMO TST, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
2004 Germany Team Selection Test, 2
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
2021 Thailand TST, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
2020 IMO Shortlist, G6
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other