📋 Table of Contents

1. Triangle Fundamentals - Basic properties, area formulas, special triangles, and midsegments
2. Triangle Centers & Special Lines - Centroid, incenter, circumcenter, and important theorems
3. Circle Properties & Theorems - Inscribed angles, cyclic quadrilaterals, and power of a point
4. Polygon Properties - Regular polygons, angle formulas, area calculations, and trapezoid midsegments
5. 3D Geometry & Solids - Volume and surface area formulas for common 3D shapes
6. Coordinate Geometry - Distance, slope, circles, and the shoelace theorem

1. Triangle Fundamentals

Essential properties and formulas for working with triangles in competition problems

1.1 Basic Triangle Properties

Triangle Area Formula For any triangle with base $b$ and height $h$: $$\text{Area} = \frac{1}{2}bh$$

Heron’s Formula For a triangle with sides $a,b,c$ and semiperimeter $s = \frac{a+b+c}{2}$: $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

Law of Sines For any triangle with sides $a,b,c$ opposite angles $A,B,C$ and circumradius $R$: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

Law of Cosines For any triangle with sides $a,b,c$ and opposite angles $A,B,C$: $$\begin{aligned} c^2 &= a^2 + b^2 - 2ab\cos C \cr b^2 &= a^2 + c^2 - 2ac\cos B \cr a^2 &= b^2 + c^2 - 2bc\cos A \end{aligned}$$

1.2 Special Right Triangles

Pythagorean Theorem For a right triangle with legs $a,b$ and hypotenuse $c$: $$c^2 = a^2 + b^2$$

30-60-90 Triangle If the short leg is $s$, then:

• Hypotenuse = $2s$
• Long leg = $s\sqrt{3}$

45-45-90 Triangle If the legs are $s$, then:

• Hypotenuse = $s\sqrt{2}$

3-4-5 Triangle Family Multiples of $(3,4,5)$: $(6,8,10)$, $(9,12,15)$, $(12,16,20)$, etc.

5-12-13 Triangle Family Multiples of $(5,12,13)$: $(10,24,26)$, $(15,36,39)$, etc.

1.3 Equilateral Triangle

For an equilateral triangle with side length $a$:

• Height: $h = \frac{\sqrt{3}}{2}a$
• Area: $\text{Area} = \frac{\sqrt{3}}{4}a^2$

1.4 The 13-14-15 Triangle

This special triangle can be split by an altitude into two right triangles with side lengths $(5,12,13)$ and $(9,12,15)$. The area is: $$\text{Area} = \frac{1}{2} \cdot 14 \cdot 12 = 84$$

Diagram (13-14-15 Triangle)

1.5 Right Triangle Properties

In a right triangle $ABC$ with $\angle B = 90^\circ$, the following triangles are similar: $$\triangle ABC \sim \triangle ADB \sim \triangle BDC$$

Altitude from $B$ to the hypotenuse $AC$ (meeting at $D$): $$BD = \frac{AB \cdot BC}{AC}$$

Further relations: $$AD \cdot CD = BD^2, \quad AD \cdot AC = AB^2, \quad CD \cdot AC = CB^2$$

Diagram (Right Triangle with Altitude)

1.6 Triangle Midsegments

Triangle Midsegment: If $D$ and $E$ are midpoints of two sides of $\triangle ABC$, then the midsegment $\overline{DE}$ is:

• Parallel to the third side: $DE \parallel BC$
• Half its length: $DE = \frac{1}{2}BC$

Diagram (Triangle Midsegment)

2. Triangle Centers & Special Lines

Understanding the key points and lines in triangles that frequently appear in competition problems

2.1 Centroid (G)

Definition: The centroid is the intersection point of the three medians of a triangle.

Key Properties:

• Each median is divided by the centroid in a 2:1 ratio (vertex to centroid : centroid to midpoint)
• The centroid is the center of mass of the triangle
• For triangle $ABC$ with centroid $G$ and midpoints $P,M,N$ of sides $AB,BC,AC$: $$AG = 2 \cdot GM, \quad BG = 2 \cdot GN, \quad CG = 2 \cdot GP$$

Diagram (Centroid with Medians)

2.2 Incenter (I)

Definition: The incenter is the intersection point of the three angle bisectors and the center of the incircle.

Key Properties:

• Equidistant from all three sides of the triangle
• The inradius $r$ is the radius of the incircle
• Area formula: $A = rs$ where $s$ is the semiperimeter

Diagram (Incenter and Incircle)

2.3 Circumcenter (O)

Definition: The circumcenter is the intersection point of the three perpendicular bisectors and the center of the circumcircle.

Key Properties:

• Equidistant from all three vertices
• The circumradius $R$ is the radius of the circumcircle
• Area formula: $A = \frac{abc}{4R}$

Diagram (Circumcenter and Circumcircle)

2.4 Ceva’s Theorem

Statement: For triangle $ABC$ with points $D \in BC$, $E \in CA$, $F \in AB$, the cevians $AD$, $BE$, $CF$ are concurrent if and only if: $$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$$

Special Cases:

• When $D,E,F$ are midpoints, the cevians are medians and intersect at the centroid
• When $D,E,F$ are feet of angle bisectors, the cevians intersect at the incenter
• When $D,E,F$ are feet of altitudes, the cevians intersect at the orthocenter

Diagram (Concurrent Cevians)

2.5 Angle Bisector Theorem

Statement: If $AD$ bisects $\angle A$ in triangle $ABC$ with $D \in \overline{BC}$, then: $$\frac{AB}{AC} = \frac{BD}{DC}$$

Diagram (Angle Bisector)

2.6 Stewart’s Theorem

Statement: Given triangle $ABC$ with sides opposite $A,B,C$ of lengths $a,b,c$. If cevian $AD$ is drawn with $BD = m$, $DC = n$, and $AD = d$, then: $$man + dad = bmb + cnc$$

Diagram (Stewart’s Theorem)

2.7 Area Ratios and Shared Angles

Shared Angle Area Ratio: When two triangles share the same angle, the ratio of their areas equals the product of the ratios of the two sides around that angle.

For triangles $ABC$ and $ADE$ sharing $\angle A$: $$\frac{[ADE]}{[ABC]} = \frac{AD \cdot AE}{AB \cdot AC}$$

Simple Area Ratio: For triangle $ABC$ with cevian $AD$ meeting $BC$ at $D$: $$\frac{BD}{DC} = \frac{[ABD]}{[ADC]}$$

This works because triangles $ABD$ and $ADC$ share the same altitude to line $BC$.

Diagram (Shared Angle Area Ratio)

3. Circle Properties & Theorems

Essential circle concepts and power relationships that are crucial for competition geometry

3.1 Inscribed Angle Theorem

Statement: The angle formed by an arc at the center (central angle) is double the angle formed by the same arc at the circumference (inscribed angle).

If the inscribed angle equals $\theta$, then the central angle equals $2\theta$.

Diagram (Inscribed Angle Theorem)

3.2 Cyclic Quadrilateral

Statement: For a quadrilateral inscribed in a circle, the sum of opposite angles is $180^\circ$: $$p + q = 180^\circ, \quad r + s = 180^\circ$$

Key Properties:

• All four vertices lie on the same circle
• Opposite angles are supplementary
• The quadrilateral has both an incircle and circumcircle

Diagram (Cyclic Quadrilateral)

3.3 Power of a Point

Two Tangents from External Point From an external point $P$ to a circle, the lengths of the tangents to the circle are equal: $$PS = PT$$

Diagram (Two Tangents)

Two Secants Intersecting Inside If chords (secants) $\overline{AB}$ and $\overline{CD}$ intersect at $P$ inside a circle with center $O$, then: $$PA \cdot PB = PC \cdot PD = r^2 - OP^2$$

Diagram (Secants Intersecting Inside)

Two Secants Intersecting Outside If secants $\overline{AB}$ and $\overline{CD}$ intersect at $P$ outside a circle with center $O$, then: $$PA \cdot PB = PC \cdot PD = OP^2 - r^2$$

Diagram (Secants Intersecting Outside)

3.4 Ptolemy’s Theorem

Statement: In a cyclic quadrilateral $ABCD$: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$

Diagram (Ptolemy’s Theorem)

3.5 Brahmagupta’s Formula

Statement: For a cyclic quadrilateral with sides $a,b,c,d$ and semiperimeter $s = \frac{a+b+c+d}{2}$, the area is: $$A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$

How to Apply:

1. Find the perimeter and divide by $2$ to get $s$
2. Compute $s-a$, $s-b$, $s-c$, $s-d$
3. Multiply the four values
4. Take the square root

3.6 Circle Sector and Arc Length

For a circle of radius $r$ and central angle $a^\circ$:

Sector Area: $$\text{Area} = \pi r^2 \cdot \frac{a^\circ}{360}$$

Arc Length: $$\text{Length} = 2\pi r \cdot \frac{a^\circ}{360}$$

Diagram (Circle Sector)

4. Polygon Properties

Understanding regular polygons, angle formulas, and area calculations

4.1 Polygon Angle Formulas

Sum of Interior Angles (n-gon) $$S_{\text{int}} = (n-2) \cdot 180^\circ$$

Interior Angle of a Regular n-gon $$\alpha_{\text{int}} = \frac{n-2}{n} \cdot 180^\circ$$

Exterior Angle of a Regular n-gon $$\alpha_{\text{ext}} = \frac{360^\circ}{n}$$

4.2 Handy Interior Angles (Regular Polygons)

4.3 Regular Hexagon Facts

• $\sum \text{ interior} = (6-2) \cdot 180^\circ = 720^\circ$
• $\alpha_{\text{int}} = \frac{6-2}{6} \cdot 180^\circ = 120^\circ$, $\alpha_{\text{ext}} = \frac{360^\circ}{6} = 60^\circ$
• Area: $A = 6 \cdot \frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2$
• Longest diagonal $= 2s$

Diagram (Regular Hexagon)

4.4 Regular Octagon Facts

• $\sum \text{ interior} = (8-2) \cdot 180^\circ = 1080^\circ$
• $\alpha_{\text{int}} = \frac{8-2}{8} \cdot 180^\circ = 135^\circ$, $\alpha_{\text{ext}} = \frac{360^\circ}{8} = 45^\circ$
• Area: $A = 2(1+\sqrt{2})s^2$

Diagram (Regular Octagon)

4.5 Quadrilateral Area Formulas

Rhombus For a rhombus with diagonals $d_1$ and $d_2$: $$\text{Area} = \frac{1}{2}d_1d_2, \quad \text{Perimeter} = 2\sqrt{d_1^2+d_2^2}$$

Diagram (Rhombus)

Trapezoid For a trapezoid with bases $b_1,b_2$ and height $h$: $$\text{Area} = \frac{b_1+b_2}{2} \cdot h$$

Diagram (Trapezoid)

4.6 Trapezoid Midsegments

Trapezoid Midsegment: In trapezoid $ABCD$ with bases $AB$ and $CD$, the segment joining the midpoints $E$ of $AD$ and $F$ of $BC$ is:

• Parallel to the bases
• Has length $EF = \frac{AB+CD}{2}$

Diagram (Trapezoid Midsegment)

5. 3D Geometry & Solids

Volume and surface area formulas for common 3D shapes encountered in competition problems

5.1 Cylinder

Volume: $V = \pi r^2 h$

Surface Area: $S = 2\pi r^2 + 2\pi r h = 2\pi r(r+h)$

Diagram (Cylinder)

5.2 Cone

Volume: $V = \frac{1}{3} \pi r^2 h$

Surface Area (with slant height $s$): $S = \pi r^2 + \pi r s = \pi r(r+s)$

Slant Height: $s = \sqrt{r^2+h^2}$

Diagram (Cone)

5.3 Sphere

Volume: $V = \frac{4}{3}\pi r^3$

Surface Area: $S = 4\pi r^2$

Diagram (Sphere)

5.4 Tetrahedron

Any Tetrahedron: $V = \frac{1}{3} (\text{base area}) \cdot (\text{height})$

Regular Tetrahedron (all edges $s$): $V = \frac{\sqrt{2}}{12}s^3$

Diagram (Tetrahedron)

5.5 Pyramid

Any Pyramid: $V = \frac{1}{3} (\text{base area}) \cdot (\text{height})$

Regular Pyramid (square base, all equal edges $s$): $V = \frac{\sqrt{2}}{6}s^3$

Diagram (Square-base Pyramid)

6. Coordinate Geometry

Essential formulas for working with points, lines, and circles in the coordinate plane

6.1 Lines and Distance

General Line: $ax + by + c = 0$

Slope-Intercept: $y = mx + b$

Slope through Two Points $(x_1,y_1),(x_2,y_2)$: $m = \frac{y_2-y_1}{x_2-x_1}$

Slope from Angle $\theta$: $m = \tan\theta$

Distance between Two Points: $d = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

Point-to-Line Distance from $(x_0,y_0)$ to $ax+by+c=0$: $D = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$

Diagram (Line through Two Points)

6.2 Circles

A circle with center $(a,b)$ and radius $r$ has equation: $$(x-a)^2+(y-b)^2=r^2$$

Diagram (Circle with Center)

6.3 Shoelace Theorem

Statement: For a simple polygon with vertices $(x_i,y_i)$ listed in order, the area is: $$A = \frac{1}{2}\left|\sum x_i y_{i+1} - \sum y_i x_{i+1}\right|$$

Key Points:

• You may list vertices in clockwise or counterclockwise order
• The name “shoelace” comes from the criss-cross multiplication pattern
• Take the absolute value of the result

Example: For vertices $A(0,0), B(4,0), C(5,2), D(2,4), E(0,3)$ in order, the computed area is $A = 15$ square units.

Diagram (Shoelace Theorem)