The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.

In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $HE$ and $FG$? [asy] defaultpen(linewidth(0.8)); size(200); pair A=(0,8), B=(10,8), C=(10,0), D=origin; pair E=(4,8), F=(10,3), G=(6,0), H=(0,5); pair I=H+4*dir(H--E); pair J=foot(I, F, G); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); draw(I--J); draw(rightanglemark(H,I,J)); draw(rightanglemark(F,J,I)); label("$A$", A, dir((5,4)--A)); label("$B$", B, dir((5,4)--B)); label("$C$", C, dir((5,4)--C)); label("$D$", D, dir((5,4)--D)); label("$E$", E, dir((5,4)--E)); label("$F$", F, dir((5,4)--F)); label("$G$", G, dir((5,4)--G)); label("$H$", H, dir((5,4)--H)); label("$d$", I--J, SW); label("3", H--A, W); label("4", E--A, N); label("6", E--B, N); label("5", F--B, dir(1)); label("3", F--C, dir(1)); label("5", H--D, W); label("4", C--G, S); label("6", D--G, S); [/asy] $ \textbf{(A)}\ 6.8\qquad\textbf{(B)}\ 7.1\qquad\textbf{(C)}\ 7.6\qquad\textbf{(D)}\ 7.8\qquad\textbf{(E)}\ 8.1 $

In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? [asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy] $\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}$

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

In the figure below, circle $O$ has two tangents, $\overline{AC}$ and $\overline{BC}$. $\overline{EF}$ is drawn tangent to circle $O$ such that $E$ is on $\overline{AC}$, $F$ is on $\overline{BC}$, and $\overline{EF} \perp \overline{FC}$. Given that the diameter of circle $O$ has length $10$ and that $CO = 13$, what is the area of triangle $EFC$? [img]https://cdn.artofproblemsolving.com/attachments/b/d/4a1bc818a5e138ae61f1f3d68f6ee5adc1ed6f.png[/img]

Let $ABCD$ be a square of side length $5$. A circle passing through $A$ is tangent to segment $CD$ at $T$ and meets $AB$ and $AD$ again at $X\ne A$ and $Y\ne A$, respectively. Given that $XY = 6$, compute $AT$.

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$. Let $\omega$ be the circle that touches the sides $AB$, $BC$, and $AD$. A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$. Determine the ratio $\frac{AK}{KD}$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Let circle $A$ have radius $9,$ and let circle $B$ have radius $5$ and be internally tangent to circle $A.$ The largest radius $r$ such that there are two circles with radius $r$ that lie inside circle $A,$ are externally tangent to each other, and externally tangent with circle $B$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Three mutually tangent spheres of radius $ 1$ rest on a horizontal plane. A sphere of radius $ 2$ rests on them. What is the distance from the plane to the top of the larger sphere? $ \textbf{(A)}\ 3 \plus{} \frac {\sqrt {30}}{2} \qquad \textbf{(B)}\ 3 \plus{} \frac {\sqrt {69}}{3} \qquad \textbf{(C)}\ 3 \plus{} \frac {\sqrt {123}}{4}\qquad \textbf{(D)}\ \frac {52}{9}\qquad \textbf{(E)}\ 3 \plus{} 2\sqrt2$

Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are collinear in that order and that $CD=DE=EF.$ What is the area of $\Gamma?$ [i]Proposed by Kyle Lee[/i]

In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Let $ABC$ be a triangle with $AB=3, BC=4, CA=5$. Let $M$ be the midpoint of $BC$, and $\Gamma$ be a circle through $A$ and $M$ that intersects $AB$ and $AC$ again at $D$ and $E$, respectively. Given that $AD=AE$, find the area of quadrilateral $MEAD$. [i]Individual #9[/i]