Practice (41)
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In triangle $ABC$, $AB = 2BC$. Given that $M$ is the midpoint of $AB$ and $\angle MCA = 60^{\circ}$, compute $\frac{CM}{AC}$ .
In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^{\circ}$. Lines $AF$, $BF$, $CE$ and $DE$ enclose a rectangle whose area is 24% of the area of $ABCD$. Compute $\frac{BF}{CE}$ .
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^{\circ}$, $\angle CDE = 168^{\circ}$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$.
Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC$ = 10 and $AD$ = 18. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$.
Let $A$ be the answer to problem 13, and let $C$ be the answer to problem 15. In the interior of angle $NOM$ = $45^{\circ}$, there is a point $P$ such that $\angle MOP = A^{\circ}$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY )^2$.
Let line segment $AB$ have length 25 and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC$ = 15, $AD$ = 24, $BC$ = 20, and $BD$ = 7. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$.
Let $AB$ be the diameter of a semicircle. $C$, $D$, and $E$ are points on $AB$, in that order, such that $AC=1$, $CD=3$, $DE=4$, and $EB=2$. From $C$ and $E$, draw perpendicular lines of $AB$, intersecting the semicircle at $F$ and $G$, respectively. Find the measurement of $\angle{FDG}$.
Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$.
Prove that triangle $PQR$ is isosceles.
(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.
Let $W_1W_2W_3$ be a triangle with circumcircle $S$, and let $A_1, A_2, A_3$ be the midpoints of $W_2W_3, W_1W_3,
W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to $S$ at $W_i$. Prove that these perpendicular lines are concurrent and identify this point of concurrency.
Let $A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that $$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$$
Given point $P_0$ in the plane of triangle $A_1A_2A_3$. Denote $A_s = A_{s-3}$, for $s > 3$. Construct points $P_1; P_2; \cdots$ sequentially such that point $P_{k+1}$ is $P_k$ rotated $120^\circ$ counter-clockwise around $A_{k+1}$. Prove that if $P_{1986} = P_0$ then triangle $A_1A_2A_3$ is isosceles.
Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the
lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.
Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.
Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.
Let $ABCD$ be a convex quadrilateral for which $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let $O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on $AB, BC, CD,$ and $DA$ respectively. Prove that lines $O_1O_3$ and $O_2O_4$ are perpendicular.
Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.
In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$. A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$. Points $C, A$, and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE=6$ and $FG=9$. Find $AG$.
In the diagram $ABCDEFG$ is a regular heptagon (a 7 sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$.
Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$.
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.