Practice (42)

back to index  |  new

Let $A$, $B$, $C$ and $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at the points $X$ and $Y$. The line $XY$ meets $BC$ at the point $Z$. Let $P$ be a point on the line $XY$ different from $Z$. The line $CP$ intersects the circle with diameter $AC$ at the points $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at the points $B$ and $N$. Prove that the lines $AM$, $DN$ and $XY$ are concurrent


(Euler Line) For any triangle $ABC$, show that the cicumcenter $O$, centroid $G$, and the orthocenter $H$ are collinear. Moreover, we have $OG:GH=1:2$.

(Euler Theorem) Let $ABC$ be a triangle, $O$, $I$ be respectively the circumcetner and incenter. Then $$OI^2 = R^2 - 2Rr$$, where $R$ denotes the circumradius and $r$ denotes the inradius.

(Nine-point circle) For any triangle $AB$C, the feet of three altitudes, the midpoints of three sides, and the midpoints of the segments from the vertices to the orthocenter, all lie on the same circle. This circle is of radius half as the circumradius of triangle $ABC$.

Show that the nine-point center $N$ lies on the Euler line, and is the middle point of $O$, the circumcenter, and $H$, the orthocenter.

Show that the triangles $ABC$, $ABH$, $BCH$, and $CAH$ all have the same nine-point circle, providing that the orthocenter $H$ does not coincide with each of the vertices $A$, $B$, $C$.

Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $R$ be the triangle's circumradius. Show that $$R = \frac{abc}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(b+a-c)}}$$

Prove the median's length formula: $$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$$

Prove the triangle's altitude formula: $$h_a = \frac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}$$

Prove the triangle's angle bisector length formula: $$t_a=\frac{2}{b+c}\sqrt{bcp(p-a)}=\sqrt{bc\Big(1-(\frac{a}{b+c})\Big)^2}$$

Triangle $ABC$ is isosceles, and $\angle{ABC}=x^\circ$. If the sum of possible measurements of $\angle{BAC}=240^\circ$, find $x^\circ$.

What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$?

(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$


What is the angle bisector's theorem?

How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?

Let $\triangle{ABC}$ be a Pythagorean triangle. If $\triangle{ABC}$'s circumstance is 30, find its circumcircle's area.

Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?

Let $\triangle{ABC}$ be a right triangle whose sides lengths are all integers. If $\triangle{ABC}$'s perimeter is 30, find its incircle's area.

In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$?

In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?


Let $\triangle{ABC}$ be a right triangle where $\angle{C} = 90^\circ$. If point $D$ is on side $BC$ or its extension, show that $$AB^2 = DB^2 + DA^2 \pm 2 \cdot DB \cdot DC$$ If $D$ is on $BC$, then the $3^{rd}$ term above takes a positive coefficient. Otherwise, if $D$ is on its extension, it takes a negative coefficient.

Let $\triangle{ABC}$ be an isosceles triangle where $AB=AC$. Show that for any point $P$ on the base $BC$ or its extension, the following relationship holds: $$AP^2 = AB\cdot AC \pm AP\cdot PB$$

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?


In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?