Practice (40)
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(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 inradius. Show that $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$
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: $$t_a=\frac{2}{b+c}\sqrt{bcp(p-a)}=\sqrt{bc\Big(1-(\frac{a}{b+c})\Big)^2}$$
Let $f(n)$ be the number of points of intersections of diagonals of a $n$-dimensional hypercube that is not the vertice of the cube. For example, $f(3) = 7$ because the intersection points of a cube's diagonals are at the centers of each face and the center of the cube. Find $f(5)$
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 $\frac{AB}{CD}$.
The sides of a right triangle all have lengths that are whole numbers. The sum of the length of one leg and the hypotenuse is 49. Find the sum of all the possible lengths of the other leg.
(A) 7 (B) 49 (C) 63 (D) 71 (E) 96
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?
In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$?
What is the smallest whole number larger than the perimeter of any triangle with a side of length $ 5$ and a side of length $19$?
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
In the diagram, AB is the diameter of a semicircle, D is the midpoint of the semicircle, angle $BAC$ is a right angle, $AC=AB$, and $E$ is the intersection of $AB$ and $CD$. Find the ratio between the areas of the two shaded regions.
What is Heron's formula to calculate a triangle's area given the lengths of three sides?
(Stewart's Theorem) Show that $$b^2m + c^2n = a(d^2 +mn)$$
What is the angle bisector's theorem?