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Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Cevians

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A Cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). There are 3 special types of cevians and their properties that we will be looking at today. 1. Median  A line joining the midpoint of a side of a triangle to the opposite vertex is called a median. Properties A median divides a triangle into two parts of equal area. $[ABD] = [ACD]$. The point where the three medians of a triangle meet is called the centroid of the triangle. Point $G$ is the centroid of $\triangle ABC$ The centroid of a triangle divides each median in the ratio $2:1$. $\frac{AG}{GD} = \frac{BG}{GE} = \frac{CG}{GF} = 2$ The three medians divide the triangle into $6$ triangles with equal area. $[AFG] = [BFG] = [BDG] = [CDG] = [CEG] = [AEG]$ An Important Result  $2(AD)^2 + 2 \cdot \big( \frac{BC}{2} \big) = (AB)^2 + (AC)^2$ 2. Angle Bisector An angle bisector of a triangle is a segment or ray that bisects an a...
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More Coordinate Geometry

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We have looked at some basics of coordinate geometry. Today, let's look at some intermediate applications of this topic. 1. Shoelace Theorem The Shoelace Theorem is an algorithm used to find the area of a polygon in the coordinate plane when the coordinates are known. If the coordinates are $(x_1, y_1), (x_2, y_2) ... (x_n, y_n), $ $\text{Area}=\frac{1}{2}\Big[(x_1y_2 - x_2y_1)+(x_2y_3 - x_3y_2) + ... + (x_ny_1 - x_1y_n)\Big].$ 2. Centroid The centroid is the point where the three medians intersect. It is also sometimes called the center of gravity for the triangle. Note: A median of a triangle is the line segment joining the vertex to the midpoint of the opposite side. If $(x_1,y_1), (x_2, y_2),$ and $(x_3,y_3)$ are the vertices of a triangle, then the coordinates of its centroid are $$\Bigg(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \Bigg).$$ 3. Incenter The incenter is the point where the three angle bisectors intersect. Note: An angle bisector of a triangle...
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Time and Work Problems

Key Facts If a person can finish a job in n days, then the work done by the person in 1 day is $\frac{1}{n}$th of the total job If a person completes $\frac{1}{n}$th of the total job in 1 day, then the time taken by the person to finish the complete job is n days.  Another version of work is pool/tank problems where there is an inlet of water and an outlet as well.   If an inlet fills a tank in n hours, then it fills $\frac{1}{n}$th part of the tank in 1 hour. i.e. work done by it in 1 hour is $\frac{1}{n}$ If an outlet empties a full tank in m hours, then it will empty $\frac{1}{m}$th part of the tank in 1 hour. i.e. work done by it is $-\frac{1}{m}$ The concept is not hard to understand, however the application can be extremely tricky. So it would only help to solve as many problems as you can on this subject. Let's look at some examples. Ex 1. A copy machine can copy a paper in $36$ minutes. If a second copy machine were to be used at the same ...
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Units Digit

Units digit of a number is the digit in the one's place of the number. For example, the units digit of $243$ is $3$. In competition math, you might come across occasional problems that ask you to find the units digit of an expression. These problems are more commonly found on Mathcounts Countdown Round. 1. The units digit of any number expressed as a power of $2$ repeats in the cycle $2, 4, 8, 6.$        $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, ....$ etc. 2. The units digit of any number expressed as a power of $3$ repeats in the cycle $3, 9, 7, 1.$        $3^1 = 3, 3^2 = 9, 4^3 = 27, 3^4 = 81, 3^5 = 243, ....$ etc. 3. The units digit of any number expressed as a power of $4$ is $4$ if the power is odd and $6$ if the power is even.        $4^1 = 4, 4^2 = 16, 4^3=64, 4^4=256, ....$ etc. 4. The units digit of any number expressed as a power of $5$ is always $5$.        $5^1...
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More Numbers

Key Facts 1. Numbers of the form $\frac{p}{q}, q \neq 0$, where $p$ and $q$ are integers and those that can be expressed in the form of terminating or repeating decimals are called rational numbers .             Ex. $\frac{7}{32} = 0.21875, \frac{8}{15} = 0.5\bar{3}$ are rational numbers. 2. Properties of operations of rational numbers     For any rational numbers $a, b, c,$     (i) Rational numbers are closed under addition, multiplication, and subtraction.          $i.e., (a+b), (a-b),$ and $(a \cdot b)$ are all rational     (ii) Rational numbers follow the commutative law  of addition and multiplication.          $i.e., a + b = b + a$ and $a \cdot b = b \cdot a.$     (iii) Rational numbers follow the associative law of addition and multiplication,          $i.e., (a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = ...
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Coordinate Geometry

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Coordinate Geometry is the study of geometry using algebraic tools like equations, relations, and operations. Before we start our deep dive, let's go over some basics of coordinate geometry. A coordinate grid consists of 2 perpendicular lines, which are called axes , and they are labeled like a number line would be. The horizontal axis is known as the x-axis , while the vertical axis is called the y-axis . The point of intersection of the two axes is known as the origin . The coordinate of the origin will always be $(0, 0)$. The numbers along the axes are used to locate points in the plane. The point at which a graph intersects the x-axis is known as the x - intercept  and the point of intersection with the y-axis is known as the y - intercept . The coordinates  also known as an Ordered Pair, is a set of values that expresses the distance a point lies from the origin. It is written in the format $(x, y)$ where $x$ is the value on the x-axis and the $y$ is the value on the ...
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Circle Theorems

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Now that you know the fundamentals of circles, let's take a look at some important circle theorems.  Inscribed angle is half the measure of the intercepted arc. An inscribed angle is the angle formed in the interior of a circle when two secant lines intersect on the circle.  The above theorem leads to an important result that comes handy in solving many geometry problems. The angle at the circumference subtended by a diameter is a right angle.  Simply put, the angle in a semicircle is a right angle.    All inscribed angles that intercept the same arc have the same measure.  The angle at the center is twice the angle at the circumference subtended by the same arc. The measure of the angle formed by the intersection of two chords is equal to the average of it's two corresponding arcs. Simply put, m$\angle \alpha = \frac{\overset{\LARGE{\frown}}{NP} + \overset{\LARGE{\frown}}{QO}}{2}$ Similar to the ...
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Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Incenter/Excenter Lemma

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Given $\triangle ABC$ with incenter $I$, extend $AI$ to meet the circumcircle of $\triangle ABC$ at $L$. Then, reflect $I$ over $L$, and name this point $I_a$. Then:  $(i)$ $BICI_a$ is a cyclic quadrilateral with point $L$ as the center of the circumscribed circle.  $(ii)$ $BI_a$ and $CI_a$ bisect the exterior angles of $\angle B$ and $\angle C$,  respectively.  Proof $(i)$ Notice that the the question is equivalent to proving the distance from $L$ to points $B, I, C, I_a$ are equal. In other words, we need to show that  $$LB = LI = LC = LI_a$$ Firstly, it is obvious that $LI = LI_a$ because by definition, $I_a$ is the reflection of $I$ over $L$, and reflections preserve the length of the segment.  Now, we are left with proving that $LB=LI$, because similar calculations will provide $LI=LC$.  We see that most of our given information involves angles, so we focus on proving $\angle LBI = \angle LIB$. Firstly,  $$\angle LBC = \angl...
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Sequences and Series

 Let's take a look at some interesting sequences and series problems. 1. (USAMTS) Evaluate the value of  $$S = \sqrt{1+\cfrac{1}{1^2}+\cfrac{1}{2^2}} + \sqrt{1+\cfrac{1}{2^2}+\cfrac{1}{3^2}} + ... + \sqrt{1+\cfrac{1}{1999^2}+\cfrac{1}{2000^2}}$$ Solution Notice that $$1+\cfrac{1}{n^2}+\cfrac{1}{(n+1)^2} = \cfrac{n^4+2n^3+3n^2+2n+1}{n^2(n+1)^2} = \cfrac{n^2+n+1)^2}{n^2(n+1)^2} = \Bigg(1+\cfrac{1}{n(n+1)} \Bigg)^2$$ Thus, $$S= \Bigg(1+\cfrac{1}{1 \cdot 2} \Bigg) + \Bigg(1+\cfrac{1}{2 \cdot 3} \Bigg) + \Bigg(1+\cfrac{1}{3 \cdot 4} \Bigg) + ... + \Bigg(1+\cfrac{1}{1999 \cdot 2000} \Bigg)$$ $$= 1999 + \Bigg(1 - \cfrac{1}{2} \Bigg) + \Bigg(\cfrac{1}{2} - \cfrac{1}{3} \Bigg) + ... + \Bigg(\cfrac{1}{1999} - \cfrac{1}{2000} \Bigg) = 1999 + 1 - \cfrac{1}{2000} = \boxed{\cfrac{3,999,999}{2000}}$$ 2. (HMMT) Find the value of  $$S = \cfrac{1}{3^2+1} + \cfrac{1}{4^2+2} + \cfrac{1}{5^2+3} + ...$$ Solution For this problem, we can use partial fraction decomposition and try to find an alt...
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