Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\bigtriangleup BXN$ is equilateral and $AC=2$. What is $BN^2$? $\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad\textbf{(B)}\ \frac{2}{9} \qquad\textbf{(C)}\ \frac{5\sqrt{2} - 3\sqrt{3}}{8} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(E)}\ \frac{3\sqrt{3} - 4}{5}$.

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$. Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$. Prove that $MN \parallel BC$ if and only if $BD=CE$.

Square $ABCD$ is divided into four rectangles by $EF$ and $GH$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $\angle BAF = 18^\circ$. $EF$ and $GH$ meet at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$. [asy] size(100); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } // NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0]; draw(A--B--C--D--cycle); draw(F--A--H); draw(E--F); draw(G--H); label("$A$",D2(A),NW); label("$B$",D2(B),SW); label("$C$",D2(C),SE); label("$D$",D2(D),NE); label("$E$",D2(E),plain.N); label("$F$",D2(F),S); label("$G$",D2(G),W); label("$H$",D2(H),plain.E); label("$P$",D2(P),SE); [/asy]

Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size. (AV Savkin)

Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are constructed on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.) a) Prove that the circles circumscribed around the outer triangles intersect in one point. b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point

Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Let $ C_1$ and $ C_2$ be circles defined by \[ (x \minus{} 10)^2 \plus{} y^2 \equal{} 36\]and \[ (x \plus{} 15)^2 \plus{} y^2 \equal{} 81,\]respectively. What is the length of the shortest line segment $ \overline{PQ}$ that is tangent to $ C_1$ at $ P$ and to $ C_2$ at $ Q$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$, find $\widehat{EMF}$.

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 $|EF|$.

Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments: $ \textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad \textbf{(D)}\ \text{is }{35}\qquad \textbf{(E)}\ \text{is }{45}$

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar? (D. Shnol)

Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.

(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles. (b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar. (A. Zaslavsky)

Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF \sim \triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.

Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$, and $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Let point $P$ be the unique point in the plane $ABC$ such that $\triangle PBM\sim\triangle PNC$. What is $AP$?

Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

In $\triangle ABC$, $D$ is on $AC$ and $F$ is on $BC$. Also, $AB \perp AC, AF \perp BC$, and $BD = DC = FC = 1$. Find $AC$. A. $\sqrt{2}$ B. $\sqrt{3}$ C. $\sqrt[3] {2}$ D. $\sqrt[3] {3}$ E. $\sqrt[4] {3}$

Let $ABC$ be a triangle with $b\neq c$. Points $D$ is the midpoint of $BC$ and let $E$ be the foot of angle $A$ bisector. In the exterior of the triangle we construct the similar triangles $AMB$ and $ANC$ . Prove: a) $MN\bot AD \Longleftrightarrow MA \bot AB$ b) $MN\bot AE \Longleftrightarrow M,A,N$ are colinear.

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles . [img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]

The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.

Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.