Points $X,Y$ lie on the side $CD$ of a convex pentagon $ABCDE$ with $X$ between $Y$ and $C$. Suppose that the triangles $\bigtriangleup XCB, \bigtriangleup ABX, \bigtriangleup AXY, \bigtriangleup AYE, \bigtriangleup YED$ are all similar (in this exact order). Prove that circumcircles of the triangles $\bigtriangleup ACD, \bigtriangleup AXY$ are tangent. [i]Pouria Mahmoudkhan Shirazi - Iran[/i]

Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$. a) Prove that $SF \perp EF.$ b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$

The point $P$, inside the triangle $[ABC]$, lies on the perpendicular bisector of $[AB]$. $Q$ and $R$ points, exterior to the triangle, they are such that $ [BPA], [BQC]$ and $[CRA]$ are similar triangles. Shows that $[PQCR]$ is a parallelogram. [img]https://cdn.artofproblemsolving.com/attachments/f/5/6e036b127f8a013794b8246cbb1544e7280d4a.png[/img]

Let $k$ be a circle with diameter $AB$. A point $C$ is chosen inside the segment $AB$ and a point $D$ is chosen on $k$ such that $BCD$ is an acute-angled triangle, with circumcentre denoted by $O$. Let $E$ be the intersection of the circle $k$ and the line $BO$ (different from $B$). Show that the triangles $BCD$ and $ECA$ are similar.

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

In a right triangle with sides $ a$ and $ b$, and hypotenuse $ c$, the altitude drawn on the hypotenuse is $ x$. Then: $ \textbf{(A)}\ ab \equal{} x^2 \qquad\textbf{(B)}\ \frac {1}{a} \plus{} \frac {1}{b} \equal{} \frac {1}{x} \qquad\textbf{(C)}\ a^2 \plus{} b^2 \equal{} 2x^2$ $ \textbf{(D)}\ \frac {1}{x^2} \equal{} \frac {1}{a^2} \plus{} \frac {1}{b^2} \qquad\textbf{(E)}\ \frac {1}{x} \equal{} \frac {b}{a}$

On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram

A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?

$\vartriangle ABC \sim \vartriangle A'B'C'$. $\vartriangle ABC$ has sides $a,b,c$ and $\vartriangle A'B'C'$ has sides $a',b',c'$. Two sides of $\vartriangle ABC$ are equal to sides of $\vartriangle A'B'C'$. Furthermore, $a < a'$, $a < b < c$, $a = 8$. Prove that there is exactly one pair of such triangles with all sides integers.

Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively. 1. Prove that triangles $IEF$ and $OKH$ are similar. 2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$. Let $ a \equal{} AB$ and $ b \equal{} CD$. For $ MN\parallel{}AB$ such that $ M$ lies on $ AD,$ $ N$ lies on $ BC$, and the trapezoids $ ABNM$ and $ MNCD$ have the same area, the length of $ MN$ equals [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number6.jpg[/img] A. $ \sqrt{ab}$ B. $ \frac{a\plus{}b}{2}$ C. $ \frac{a^2 \plus{} b^2}{a\plus{}b}$ D. $ \sqrt{\frac{a^2 \plus{} b^2}{2}}$ E. $ \frac{a^2 \plus{} (2 \sqrt{2} \minus{} 2)ab \plus{} b^2}{\sqrt{2} (a\plus{}b)}$

Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

Let $ K$ be a point inside the triangle $ ABC$. Let $ M$ and $ N$ be points such that $ M$ and $ K$ are on opposite sides of the line $ AB$, and $ N$ and $ K$ are on opposite sides of the line $ BC$. Assume that $ \angle MAB \equal{} \angle MBA \equal{} \angle NBC \equal{} \angle NCB \equal{} \angle KAC \equal{} \angle KCA$. Show that $ MBNK$ is a parallelogram.

In triangle $ABC$, $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular from $A$ to $BC$, and $F$ is the foot of the perpendicular from $D$ to $AC$. Given that $BE=5$, $EC=9$, and the area of triangle $ABC$ is $84$, compute $|EF|$.

For an isosceles triangle $ABC$ where $AB=AC$, it is possible to construct, using only compass and straightedge, an isosceles triangle $PQR$ where $PQ=PR$ such that triangle $PQR$ is similar to triangle $ABC$, point $P$ is in the interior of line segment $AC$, point $Q$ is in the interior of line segment $AB$, and point $R$ is in the interior of line segment $BC$. Describe one method of performing such a construction. Your method should work on every isosceles triangle $ABC$, except that you may choose an upper limit or lower limit on the size of angle $BAC$. [asy] defaultpen(linewidth(0.7)); pair a= (79,164),b=(19,22),c=(138,22),p=(109,91),q=(38,67),r=(78,22); pair point = ((p.x+q.x+r.x)/3,(p.y+q.y+r.y)/3); draw(a--b--c--cycle); draw(p--q--r--cycle); label("$A$",a,dir(point--a)); label("$B$",b,dir(point--b)); label("$C$",c,dir(point--c)); label("$P$",p,dir(point--p)); label("$Q$",q,dir(point--q)); label("$R$",r,dir(point--r));[/asy]

Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP,BP$ and $CP$ intersect circle $\omega$ for the second time at $D,E$ and $F,$ respectively. If $A',B',C'$ are the reflections of $A,B,C$ with respect to the lines $EF,FD,DE,$ respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.

Let $P$ be a point on the interior of triangle $ABC$, such that the triangle $ABP$ satisfies $AP = BP$. On each of the other sides of $ABC$, build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$ and $$CR = RA.$$ Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

Let $ABC$ and $AMN$ be two similar triangles with the same orientation, such that $AB=AC$, $AM=AN$ and having disjoint interiors. Let $O$ be the circumcenter of the triangle $MAB$. Prove that the points $O$, $C$, $N$, $A$ lie on the same circle if and only if the triangle $ABC$ is equilateral. [i]Valentin Vornicu[/i]

Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.

a.) Prove that for $n = 4$ or $n \geq 6$ each triangle $ABC$ can be decomposed in $n$ similar (not necessarily congruent) triangles. b.) Show: An equilateral triangle can neither be composed in 3 nor 5 triangles. c.) Is there a triangle $ABC$ which can be decomposed in 3 and 5 triangles, analogously to a.). Either give an example or prove that there is not such a triangle.