Found problems: 410
Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.
Let $E$ be a point on side $[AD]$ of rhombus $ABCD$. Lines $AB$ and $CE$ meet at $F$, lines $BE$ and $DF$ meet at $G$. If $m(\widehat{DAB}) = 60^\circ $, what is$m(\widehat{DGB})$?
$
\textbf{a)}\ 45^\circ
\qquad\textbf{b)}\ 50^\circ
\qquad\textbf{c)}\ 60^\circ
\qquad\textbf{d)}\ 65^\circ
\qquad\textbf{e)}\ 75^\circ
$
Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$.
Show that $SO \perp XY$.
Point $ E$ is selected on side $ AB$ of triangle $ ABC$ in such a way that $ AE: EB \equal{} 1: 3$ and point $ D$ is selected on side $ BC$ such that $ CD: DB \equal{} 1: 2$. The point of intersection of $ AD$ and $ CE$ is $ F$. Then $ \frac {EF}{FC} \plus{} \frac {AF}{FD}$ is:
$ \textbf{(A)}\ \frac {4}{5} \qquad \textbf{(B)}\ \frac {5}{4} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \frac {5}{2}$
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that
\[
\angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ}
\]
if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
[i]Proposed by Dusan Djukic, Serbia[/i]
Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.
The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.
A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.
Let $P$ be an interior point of a triangle $ABC$ and $AP,BP,CP$ meet the sides $BC,CA,AB$ in $D,E,F$ respectively. Show that \[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]
[hide="Remark"]This is known as [i]Van Aubel's[/i] Theorem.[/hide]
Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.
$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that
$$ TB\cdot TC=TI_b^2.$$
Show that it is not possible to have a triangle with sides $a,b,$ and $c$ whose medians have length $\frac{2}{3}a, \frac{2}{3}b$ and $\frac{4}{5}c$.
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$.
[i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]
In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$.
Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.
Let $ ABCDE$ be a convex pentagon such that
\[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE.
\]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.
[i]Proposed by Zuming Feng, USA[/i]
Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$.
Nguyễn Minh Hà
$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy]
Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY=\angle CBY$ and $\overline{BE}\perp\overline{AC}$. Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ=\angle BCZ$ and $\overline{CF}\perp\overline{AB}$.
Lines $\overleftrightarrow{I_BF}$ and $\overleftrightarrow{I_CE}$ meet at $P$. Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.
[i]Proposed by Evan Chen and Telv Cohl[/i]
Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly.
1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$.
2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$, projections of $E$ onto $CA$ and $CB$ are $R$ and $S$. Let $AP \cap CQ=X$, $AS \cap BR=Y$ and $BX \cap CY=Z$. Show that $AZ \perp BC$.