Practice (41)

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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.


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


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 $c=AB$.

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.


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


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


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 $AB$.


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 $PB$.


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


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

What is the Ptolemy Theorem?

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.

As shown, points $X$ and $Y$ are on the extension of $BC$ in $\triangle{ABC}$ such that the order of these four points are $X$, $B$, $C$, and $Y$. Meanwhile, they satisfy the relation $BX\cdot AC = CY\cdot AB$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle{ACX}$ and $\triangle{ABY}$, respectively. If $O_1O_2$ intersects $AB$ and $AC$ at $U$ and $V$, respectively, show that $\triangle{AUV}$ is isosceles.


(British Flag Theorem) Let point $P$ lie inside rectangle $ABCD$. Draw four squares using each of $AP$, $BP$, $CP$, and $DP$ as one side. Show that $$S_{AA_1A_2P}+S_{CC_1C_2P}=S_{BB_1B_2P}+S_{DD_1D_2P}$$


(De Gua's Theorem) In a trirectangular tetrahedron $ABCD$ where $A$ is the shared right-angle corner. Show that $$S_{\triangle{BCD}}^2=S_{\triangle{ABC}}^2+S_{\triangle{ACD}}^2+S_{\triangle{ADB}}^2$$

Let $CD$ be the altitude in right $\triangle{ABC}$ from the right angle $C$. If inradii of $\triangle{ABC}$, $\triangle{ACD}$, and $\triangle{BCD}$ be $r_1$, $r_2$, and $r_3$, respectively, show that $$r_1 + r_2 + r_3 = CD$$

Three circles are tangent to each other and also a common line, as shown. Let the radii of circles $O_1$, $O_2$, and $O_3$ be $r_1$, $r_2$, and $r_3$, respectively. Show that $$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_1}} +\frac{1}{\sqrt{r_2}}$$


Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$

Let point $P$ inside an equilateral $\triangle{ABC}$ such that $AP=3$, $BP=4$, and $CP=5$. Find the side length of $\triangle{ABC}$.

Let $M$ be a point inside $\triangle{ABC}$. Draw $MA'\perp BC$, $MB'\perp CA$, and $MC'\perp AB$ such that $BA'=BC'$ and $CA'=CB'$. Prove $AB'=AC'$.


Four sides of a concyclic quadrilateral have lengths of 25, 39, 52, and 60, in that order. Find the circumference of its circumcircle.

In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that $$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$

Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.

A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\tfrac{x}{y}$?

Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{4}$ and $\frac{BY}{BD} = \frac{11}{36}$. Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\cdot XG$?