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 angle bisector theorem
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 $x^\circ$.
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$. If $IA=5$, $IB=7$, $IC=4$, $ID=9$, find the value of $\frac{AB}{CD}$.
There are $n$ circles inside a square $ABC$ whose side's length is $a$. If the area of any circle is no more than 1, and every line that is parallel to one side of $ABCD$ intersects at most one such circle, show that the sum of the area of all these $n$ circles is less than $a$.
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?
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)?
Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?
Let $\triangle{ABC}$ be a Pythagorean triangle. If $\triangle{ABC}$'s circumstance is 30, find its circumcircle's area.
In triangle $ABC$ , side $AC$ and the perpendicular bisector of $BC$ meet in point $D$, and $BD$ bisects $\angle ABC$. If $AD=9$ and $DC=7$, what is the area of triangle ABD?
Find the degree measure of an angle whose complement is 25% of its supplement.
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?
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 $Q$?
In the diagram , $PA = QB = PC = QC = PD = QD = 1, CE = CF = EF$ and $EA = BF = 2AB$. Determine $BD$.
In rectangle $ABCD$, we have $AB=8$, $BC=9$, $H$ is on $BC$ with $BH=6$, $E$ is on $AD$ with $DE=4$, line $EC$ intersects line $AH$ at $G$, and $F$ is on line $AD$ with $GF \perp AF$. Find the length of $GF$.
A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
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 $r+s+t$?