Practice (40)

3107

As shown, $E, F, G, H$ are midpoints of the four sides. If $AB\parallel A'B', BC\parallel B'C',$ and $CD\parallel C'D'$, show that $AD\parallel A'D'$.

3108

As shown.

3109

As shown, two vertices of a square are on the circle and one side is tangent to the circle. If the side length of the square is 8, find the radius of the circle.

3111

Triangle $ABC$ is isosceles. An ant begins at $A$, walks exactly halfway along the perimeter of $\triangle{ABC}$, and then returns directly $A$, cutting through the interior of the triangle. The ant's path surround exactly 90% of the area of $\triangle{ABC}$. Compute the maximum value of $\tan{A}$.

3120

In $\triangle{ABC}$, $\angle{BAC} = 40^\circ$ and $\angle{ABC} = 60^\circ$. Points $D$ and $E$ are on sides $AC$ and $AB$, respectively, such that $\angle{DBC}=40^\circ$ and $\angle{ECB}=70^\circ$. Let $F$ be the intersection point of $BD$ and $CE$. Show that $AF\perp BC$.

3121

$DEB$ is a chord of a circle such that $DE=3$ and $EB=5 .$ Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C.$ Given $EC=1,$ find the radius of the circle

3122

Chords $AB$ and $CD$ of a given circle are perpendicular to each other and intersect at a right angle at point $E$. If $BE=16$, $DE=4$, and $AD=5$, find $CE$.

3131

See the attachments

3135

Let $O$ be a point inside a convex pentagon, as shown, such that $\angle{1} = \angle{2}, \angle{3} = \angle{4}, \angle{5} = \angle{6},$ and $\angle{7} = \angle{8}$. Show that either $\angle{9} = \angle{10}$ or $\angle{9} + \angle{10} = 180^\circ$ holds.

3137

Let quadrilateral $ABCD$ inscribe a circle. If $BE=ED$, prove $$AB^2+BC^2 +CD^2 + DA^2 = 2AC^2$$

3138

Let $H$ be the orthocenter of acute $\triangle{ABC}$. Show that $$a\cdot BH\cdot CH + b\cdot CH\cdot AH+c\cdot AH\cdot BH=abc$$ where $a=BC, b=CA,$ and $c=AB$.

3139

In $\triangle{ABC}$, $AE$ and $AF$ trisects $\angle{A}$, $BF$ and $BD$ trisects $\angle{B}$, $CD$ and $CE$ trisects $\angle{C}$. Show that $\triangle{DEF}$ is equilateral.

3140

As shown, $\angle{ACB} = 90^\circ$, $AD=DB$, $DE=DC$, $EM\perp AB$, and $EN\perp CD$. Prove $$MN\cdot AB = AC\cdot CB$$

3141

As shown, in $\triangle{ABC}$, $AB=AC$, $\angle{A} = 20^\circ$, $\angle{ABE} = 30^\circ$, and $\angle{ACD}=20^\circ$. Find the measurement of $\angle{CDE}$.

3142

Let $O_1$ and $O_2$ be two intersecting circles. Let a common tangent to these two circles touch $O_1$ at $A$ and $O_2$ at $B$. Show that the common chord of these two circles, when extended, bisects segment $AB$.

3143

Line $PT$ is tangent to circle $O$ at point $T$. $PA$ intersects circle $O$ and its diameter $CT$ at $B$, $D$, and $A$ in that order. If $CD=2, AD=3,$ and $BD=6$, find the length of $PB$.

3144

In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.

3145

In tetrahedron $P-ABC$, $AB=BC=CA$ and $PA=PB=PC$. If $AB=1$ and the altitude from $P$ to $ABC$ is $\sqrt{2}$, find the radius of $P-ABC$'s inscribed sphere.

3146

Let $AB=2$ is a diameter of circle $O$. If $AC=AO$, $AC\perp AB$, $BD=\frac{3}{2}\cdot AB$, $BD\perp AB$ and $P$ is a point on arc $AB$. Find the largest possible area of the enclosed polygon $ABDPC$.

3147

Two circles, $O_1$ and $O_2$ are tangent. Let $AB$ be their common tangent line which touches $O_1$ at point $A$ and touches $O_2$ at point $B$. Extend $AO_1$ and intersects $O_1$ at another point $C$. Line $CD$ is tangent to circle $O_2$ at point $D$. Show that $AC=CD$.

3148

Let $P$ be a point inside square $ABCD$ such that $AP=1, BP = 3,$ nd $DP=\sqrt{7}$. Find the area of $ABCD$. Try to find at least two solutions.

3149

Let $M$ be the midpoint of $AB$ which is the hypotenuse of a non-isosceles right triangle ${ABC}$. If $DM\perp AB$ and $DC$ bisects $\angle{ACB}$, show $CM=DM$.

3150

Let $ABCD$ be a square. Point $E$ is outside $ABCD$ such that $DE=CD$ and $\angle{AED}=15^\circ$. Show $\triangle{DCE}$ is equilateral.

3166

What is the Ptolemy Theorem?

3207

Show that if $a, b, c$ are the lengths of the sides of a triangle, then the equation $$b^2x^2 +(b^2+c^2-a^2)x+c^2=0$$ does not have real roots.

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