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This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

You have probably been wondering whether our powers of algebraic problem solving break down if we divide by the variable or we have entire expressions in denominator of a fraction. Well, they don't! In this topic, you'll learn how to interpret and manipulate rational expressions (when you have one algebraic expression divided by another)!. Simplifying Rational Expressions Introduction. Simplifying Rational Expressions 1. Dividing polynomials by binomials 1. Simplifying Rational Expressions 2. Dividing polynomials by binomials 2. Simplifying rational expressions 2. Simplifying Rational Expressions 3. Simplifying Rational Expressions Example 2. Dividing polynomials by binomials 3. Simplifying rational expressions 3. Simplifying rational expressions 4. Adding and Subtracting Rational Expressions. Adding and Subtracting Rational Expressions 2. Adding and Subtracting Rational Expressions 3. Subtracting Rational Expressions. Simplifying First for Subtracting Rational Expressions. Adding and subtracting rational expressions 0.5. Adding and subtracting rational expressions 1. Adding and subtracting rational expressions 1.5. Adding and subtracting rational expressions 2. Adding and subtracting rational expressions 3. Multiplying and Simplifying Rational Expressions. Multiplying and Dividing Rational Expressions 1. Multiplying and Dividing Rational Expressions 2. Multiplying and Dividing Rational Expressions 3. Multiplying and dividing rational expressions 1. Multiplying and dividing rational expressions 2. Multiplying and dividing rational expressions 3. Multiplying and dividing rational expressions 4. Multiplying and dividing rational expressions 5. Rationalizing Denominators of Expressions. Asymptotes of Rational Functions. Another Rational Function Graph Example. A Third Example of Graphing a Rational Function. Partial Fraction Expansion 1. Partial Fraction Expansion 2. Partial Fraction Expansion 3. Partial fraction expansion. Ex 1 Multi step equation. Solving rational equations 1. Rational Equations. Solving Rational Equations 1. Solving Rational Equations 2. Solving Rational Equations 3. Applying Rational Equations 1. Applying Rational Equations 2. Applying Rational Equations 3. Solving rational equations 2. Extraneous Solutions to Rational Equations. Extraneous solutions. Rational Inequalities. Rational Inequalities 2. Simplifying Rational Expressions Introduction. Simplifying Rational Expressions 1. Dividing polynomials by binomials 1. Simplifying Rational Expressions 2. Dividing polynomials by binomials 2. Simplifying rational expressions 2. Simplifying Rational Expressions 3. Simplifying Rational Expressions Example 2. Dividing polynomials by binomials 3. Simplifying rational expressions 3. Simplifying rational expressions 4. Adding and Subtracting Rational Expressions. Adding and Subtracting Rational Expressions 2. Adding and Subtracting Rational Expressions 3. Subtracting Rational Expressions. Simplifying First for Subtracting Rational Expressions. Adding and subtracting rational expressions 0.5. Adding and subtracting rational expressions 1. Adding and subtracting rational expressions 1.5. Adding and subtracting rational expressions 2. Adding and subtracting rational expressions 3. Multiplying and Simplifying Rational Expressions. Multiplying and Dividing Rational Expressions 1. Multiplying and Dividing Rational Expressions 2. Multiplying and Dividing Rational Expressions 3. Multiplying and dividing rational expressions 1. Multiplying and dividing rational expressions 2. Multiplying and dividing rational expressions 3. Multiplying and dividing rational expressions 4. Multiplying and dividing rational expressions 5. Rationalizing Denominators of Expressions. Asymptotes of Rational Functions. Another Rational Function Graph Example. A Third Example of Graphing a Rational Function. Partial Fraction Expansion 1. Partial Fraction Expansion 2. Partial Fraction Expansion 3. Partial fraction expansion. Ex 1 Multi step equation. Solving rational equations 1. Rational Equations. Solving Rational Equations 1. Solving Rational Equations 2. Solving Rational Equations 3. Applying Rational Equations 1. Applying Rational Equations 2. Applying Rational Equations 3. Solving rational equations 2. Extraneous Solutions to Rational Equations. Extraneous solutions. Rational Inequalities. Rational Inequalities 2.

Extending our understanding of numbers below 0. Thinking about "absolute" distance from 0. Negative Numbers Introduction. Negative numbers on the number line. Number line 3. Ordering Negative Numbers. Ordering negative numbers. Number opposites. Number opposites. Negative Numbers Introduction. Number opposites. Example: Adding negative numbers. Example: Adding integers with different signs. Adding negative numbers. Adding and subtracting negative number examples. Why Subtracting a Negative Equivalent to Adding a Positive. Adding and subtracting negative numbers. Negative number word problems 1. Negative number word problems. Adding decimals of different signs word problem. Adding/Subtracting negative numbers. Multiplying Positive and Negative Numbers. Why a Negative Times a Negative is a Positive. Why a Negative Times a Negative Makes Intuitive Sense. Dividing Positive and Negative Numbers. Example: Multiplying numbers with different signs. Multiplying and dividing negative numbers. Multiplying and dividing negative numbers. Absolute Value and Number Lines. Absolute Value 1. Finding absolute values. Absolute Value of Integers. Comparing Absolute Values. Comparing absolute values. Absolute value word problems exercise. Absolute value word problems. Constructing and interpreting absolute value. Constructing and interpreting absolute value. Negative Numbers Introduction. Negative numbers on the number line. Number line 3. Ordering Negative Numbers. Ordering negative numbers. Number opposites. Number opposites. Negative Numbers Introduction. Number opposites. Example: Adding negative numbers. Example: Adding integers with different signs. Adding negative numbers. Adding and subtracting negative number examples. Why Subtracting a Negative Equivalent to Adding a Positive. Adding and subtracting negative numbers. Negative number word problems 1. Negative number word problems. Adding decimals of different signs word problem. Adding/Subtracting negative numbers. Multiplying Positive and Negative Numbers. Why a Negative Times a Negative is a Positive. Why a Negative Times a Negative Makes Intuitive Sense. Dividing Positive and Negative Numbers. Example: Multiplying numbers with different signs. Multiplying and dividing negative numbers. Multiplying and dividing negative numbers. Absolute Value and Number Lines. Absolute Value 1. Finding absolute values. Absolute Value of Integers. Comparing Absolute Values. Comparing absolute values. Absolute value word problems exercise. Absolute value word problems. Constructing and interpreting absolute value. Constructing and interpreting absolute value.

Understanding and solving exponents without algebra. Introduction to exponents. Raising a number to the 0th and 1st power. Powers of 1 and -1. Powers of fractions. Powers of zero. Exponent example 1. Exponent example 2. Positive and zero exponents. Understanding Square Roots. Square roots of perfect squares. Approximating Square Roots. Estimating square roots. Simplifying square roots. Simplifying radicals. Square Roots and Real Numbers. Simplifying radicals. Simplifying radicals 2. Adding and simplifying radicals. Adding and subtracting radicals. Finding Cube Roots. Cube root of a non-perfect cube. Simplifying a cube root. Cube roots. Cube roots 2. Patterns in zeros exercise. Patterns in zeros. Exponent Rules Part 1. Exponent Rules Part 2. Exponent Properties Involving Quotients. Exponent rules. Negative exponents. Negative Exponent Intuition. Zero, Negative, and Fractional Exponents. Negative exponents. Basic fractional exponents. Negative fractional exponent examples. Negative fractional exponent examples 2. Fractional exponents. Fractional exponents with numerators other than 1. Fractional exponents 2. Introduction to scientific notation. Scientific Notation. Scientific notation intuition. Scientific Notation Examples. Scientific Notation I. Scientific Notation Example 2. Scientific notation. Orders of magnitude exercise example 1. Orders of magnitude exercise example 2. Orders of magnitude. Multiplying in Scientific Notation. Multiplying in scientific notation example. Dividing in scientific notation example. Multiplying and dividing in scientific notation. Multiplying and dividing scientific notation. Simplifying a complicated expression into scientific notation. Calculating red blood cells in the body using scientific notation. Computing in scientific notation. Introduction to exponents. Raising a number to the 0th and 1st power. Powers of 1 and -1. Powers of fractions. Powers of zero. Exponent example 1. Exponent example 2. Positive and zero exponents. Understanding Square Roots. Square roots of perfect squares. Approximating Square Roots. Estimating square roots. Simplifying square roots. Simplifying radicals. Square Roots and Real Numbers. Simplifying radicals. Simplifying radicals 2. Adding and simplifying radicals. Adding and subtracting radicals. Finding Cube Roots. Cube root of a non-perfect cube. Simplifying a cube root. Cube roots. Cube roots 2. Patterns in zeros exercise. Patterns in zeros. Exponent Rules Part 1. Exponent Rules Part 2. Exponent Properties Involving Quotients. Exponent rules. Negative exponents. Negative Exponent Intuition. Zero, Negative, and Fractional Exponents. Negative exponents. Basic fractional exponents. Negative fractional exponent examples. Negative fractional exponent examples 2. Fractional exponents. Fractional exponents with numerators other than 1. Fractional exponents 2. Introduction to scientific notation. Scientific Notation. Scientific notation intuition. Scientific Notation Examples. Scientific Notation I. Scientific Notation Example 2. Scientific notation. Orders of magnitude exercise example 1. Orders of magnitude exercise example 2. Orders of magnitude. Multiplying in Scientific Notation. Multiplying in scientific notation example. Dividing in scientific notation example. Multiplying and dividing in scientific notation. Multiplying and dividing scientific notation. Simplifying a complicated expression into scientific notation. Calculating red blood cells in the body using scientific notation. Computing in scientific notation.

Introduction to Ratios (new HD version). Ratios as Fractions in Simplest Form. Simplifying Rates and Ratios. Expressing ratios as fractions. Describing ratios exercise. Describing ratios. Ratio word problem exercise example 1. Ratio word problem exercise example 2. Ratio word problems. Proportion validity example. Solving ratio problems with tables exercise. Solving ratio problems with tables exercise 2. Solving ratio problems with tables exercise 3. Solving ratio problems with tables. Usain Bolt's Average Speed. Finding Unit Rates. Finding Unit Prices. Rate problems 0.5. Multiple rates word problem. Rate problems 2. Converting pounds to ounces. Converting Gallons to quarts pints and cups. Converting Yards into Inches. Performing arithmetic calculations on units of volume. Application problems involving units of weight. Solving application problems involving units of volume. Gas cost for a roadtrip. Squirrel survival. Converting units of length. Unit Conversion Example: Drug Dosage. Unit Conversion with Fractions. Units. Speed translation. Unit conversion within the metric system. Converting within the metric system. Perimeter and Unit Conversion. Applying the Metric System. Conversion between metric units. U.S. Customary and Metric units. Converting Fahrenheit to Celsius. Comparing Celsius and Farenheit temperature scales. Introduction to Ratios (new HD version). Ratios as Fractions in Simplest Form. Simplifying Rates and Ratios. Expressing ratios as fractions. Describing ratios exercise. Describing ratios. Ratio word problem exercise example 1. Ratio word problem exercise example 2. Ratio word problems. Proportion validity example. Solving ratio problems with tables exercise. Solving ratio problems with tables exercise 2. Solving ratio problems with tables exercise 3. Solving ratio problems with tables. Usain Bolt's Average Speed. Finding Unit Rates. Finding Unit Prices. Rate problems 0.5. Multiple rates word problem. Rate problems 2. Converting pounds to ounces. Converting Gallons to quarts pints and cups. Converting Yards into Inches. Performing arithmetic calculations on units of volume. Application problems involving units of weight. Solving application problems involving units of volume. Gas cost for a roadtrip. Squirrel survival. Converting units of length. Unit Conversion Example: Drug Dosage. Unit Conversion with Fractions. Units. Speed translation. Unit conversion within the metric system. Converting within the metric system. Perimeter and Unit Conversion. Applying the Metric System. Conversion between metric units. U.S. Customary and Metric units. Converting Fahrenheit to Celsius. Comparing Celsius and Farenheit temperature scales.

This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.

Random logic puzzles and brain teasers. Fun to do and useful for many job interviews!. Liar Truth-teller Brain Teaser. Toggler Brain Teaser. Alien Abduction Brain Teaser. Blue Forehead Room Brain Teaser. Blue Forehead Room Solution. Forehead Numbers Brain Teaser. Light Bulb Switching Brain Teaser. Path Counting Brain Teaser. 3-D Path Counting Brain Teaser. Liar Truth-teller Brain Teaser. Toggler Brain Teaser. Alien Abduction Brain Teaser. Blue Forehead Room Brain Teaser. Blue Forehead Room Solution. Forehead Numbers Brain Teaser. Light Bulb Switching Brain Teaser. Path Counting Brain Teaser. 3-D Path Counting Brain Teaser.

Sal working through the 53 problems from the practice test available at http://www.cde.ca.gov/ta/tg/hs/documents/mathpractest.pdf for the CAHSEE (California High School Exit Examination). Clearly useful if you're looking to take that exam. Probably still useful if you want to make sure you have a solid understanding of basic high school math. CAHSEE Practice: Problems 1-3. CAHSEE Practice: Problems 4-9. CAHSEE Practice: Problems 10-12. CAHSEE Practice: Problems 13-14. CAHSEE Practice: Problems 15-16. CAHSEE Practice: Problems 17-19. CAHSEE Practice: Problems 20-22. CAHSEE Practice: Problems 23-27. CAHSEE Practice: Problems 28-31. CAHSEE Practice: Problems 32-34. CAHSEE Practice: Problems 35-37. CAHSEE Practice: Problems 38-42. CAHSEE Practice: Problems 43-46. CAHSEE Practice: Problems 47-51. CAHSEE Practice: Problems 52-53. CAHSEE Practice: Problems 1-3. CAHSEE Practice: Problems 4-9. CAHSEE Practice: Problems 10-12. CAHSEE Practice: Problems 13-14. CAHSEE Practice: Problems 15-16. CAHSEE Practice: Problems 17-19. CAHSEE Practice: Problems 20-22. CAHSEE Practice: Problems 23-27. CAHSEE Practice: Problems 28-31. CAHSEE Practice: Problems 32-34. CAHSEE Practice: Problems 35-37. CAHSEE Practice: Problems 38-42. CAHSEE Practice: Problems 43-46. CAHSEE Practice: Problems 47-51. CAHSEE Practice: Problems 52-53.

Example problems from random math competitions. 2003 AIME II Problem 1. 2003 AIME II Problem 3. 2003 AIME II Problem 4 (part 1). Sum of factors of 27000. Sum of factors 2. 2003 AIME II Problem 5. 2003 AIME II Problem 5 Minor Correction. Area Circumradius Formula Proof. 2003 AIME II Problem 8. Sum of Polynomial Roots (Proof). Sum of Squares of Polynomial Roots. 2003 AIME II Problem 9. 2003 AIME II Problem 12. 2003 AIME II Problem 13. 2003 AIME II Problem 10. 2003 AIME II Problem 11. 2003 AIME II Problem 14. 2003 AIME II Problem 15 (part 1). 2003 AIME II Problem 15 (part 2). 2003 AIME II Problem 15 (part 3). 2003 AIME II Problem 1. 2003 AIME II Problem 3. 2003 AIME II Problem 4 (part 1). Sum of factors of 27000. Sum of factors 2. 2003 AIME II Problem 5. 2003 AIME II Problem 5 Minor Correction. Area Circumradius Formula Proof. 2003 AIME II Problem 8. Sum of Polynomial Roots (Proof). Sum of Squares of Polynomial Roots. 2003 AIME II Problem 9. 2003 AIME II Problem 12. 2003 AIME II Problem 13. 2003 AIME II Problem 10. 2003 AIME II Problem 11. 2003 AIME II Problem 14. 2003 AIME II Problem 15 (part 1). 2003 AIME II Problem 15 (part 2). 2003 AIME II Problem 15 (part 3).

If you can take one figure and flip, shift and rotate (not resize) it to be identical to another figure, then the two figures are congruent. This topic explores this foundational idea in geometry. Congruent Triangles and SSS. SSS to Show a Radius is Perpendicular to a Chord that it Bisects. Other Triangle Congruence Postulates. Two column proof showing segments are perpendicular. Finding Congruent Triangles. Congruency postulates. More on why SSA is not a postulate. Perpendicular Radius Bisects Chord. Congruent Triangle Proof Example. Congruent Triangle Example 2. Congruent triangles 1. Congruent triangles 2. Congruent legs and base angles of Isosceles Triangles. Equilateral Triangle Sides and Angles Congruent. Equilateral and Isosceles Example Problems. Triangle types. Triangle angles 1. Another Isosceles Example Problem. Example involving an isosceles triangle and parallel lines. Figuring out all the angles for congruent triangles example. Basic Triangle Proofs Module Example. Basic Triangle Proofs Module Example 2. Basic triangle proofs. Fill-in-the-blank triangle proofs example 1. Fill-in-the-blank triangle proofs example 2. Fill-in-the-blank triangle proofs. Wrong statements in triangle proofs example 1. Wrong statements in triangle proofs. Problem involving angle derived from square and circle. Congruent Triangles and SSS. SSS to Show a Radius is Perpendicular to a Chord that it Bisects. Other Triangle Congruence Postulates. Two column proof showing segments are perpendicular. Finding Congruent Triangles. Congruency postulates. More on why SSA is not a postulate. Perpendicular Radius Bisects Chord. Congruent Triangle Proof Example. Congruent Triangle Example 2. Congruent triangles 1. Congruent triangles 2. Congruent legs and base angles of Isosceles Triangles. Equilateral Triangle Sides and Angles Congruent. Equilateral and Isosceles Example Problems. Triangle types. Triangle angles 1. Another Isosceles Example Problem. Example involving an isosceles triangle and parallel lines. Figuring out all the angles for congruent triangles example. Basic Triangle Proofs Module Example. Basic Triangle Proofs Module Example 2. Basic triangle proofs. Fill-in-the-blank triangle proofs example 1. Fill-in-the-blank triangle proofs example 2. Fill-in-the-blank triangle proofs. Wrong statements in triangle proofs example 1. Wrong statements in triangle proofs. Problem involving angle derived from square and circle.

This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.

Acknowledgement

Prof. McKernan would like to acknowledge the contributions of Lars Hesselholt to the development of this course.

This topic continues our journey through the world of Euclid by helping us understand angles and how they can relate to each other. Angle basics. Measuring angles in degrees. Using a protractor. Measuring angles. Measuring angles. Acute right and obtuse angles. Angle types. Vertical, adjacent and linearly paired angles. Exploring angle pairs. Introduction to vertical angles. Vertical angles. Using algebra to find the measures of vertical angles. Vertical angles 2. Proof-Vertical Angles are Equal. Angles Formed by Parallel Lines and Transversals. Identifying Parallel and Perpendicular Lines. Figuring out angles between transversal and parallel lines. Congruent angles. Parallel lines 1. Using algebra to find measures of angles formed from transversal. Parallel lines 2. CA Geometry: Deducing Angle Measures. Proof - Sum of Measures of Angles in a Triangle are 180. Triangle Angle Example 1. Triangle Angle Example 2. Triangle Angle Example 3. Challenging Triangle Angle Problem. Proof - Corresponding Angle Equivalence Implies Parallel Lines. Finding more angles. Angles 1. Angles 2. Sum of Interior Angles of a Polygon. Angles of a polygon. Sum of the exterior angles of convex polygon. Introduction to angles (old). Angles (part 2). Angles (part 3). Angles formed between transversals and parallel lines. Angles of parallel lines 2. The Angle Game. Angle Game (part 2). Acute right and obtuse angles. Complementary and supplementary angles. Complementary and supplementary angles. Example using algebra to find measure of complementary angles. Example using algebra to find measure of supplementary angles. Angle addition postulate. Angle basics. Measuring angles in degrees. Using a protractor. Measuring angles. Measuring angles. Acute right and obtuse angles. Angle types. Vertical, adjacent and linearly paired angles. Exploring angle pairs. Introduction to vertical angles. Vertical angles. Using algebra to find the measures of vertical angles. Vertical angles 2. Proof-Vertical Angles are Equal. Angles Formed by Parallel Lines and Transversals. Identifying Parallel and Perpendicular Lines. Figuring out angles between transversal and parallel lines. Congruent angles. Parallel lines 1. Using algebra to find measures of angles formed from transversal. Parallel lines 2. CA Geometry: Deducing Angle Measures. Proof - Sum of Measures of Angles in a Triangle are 180. Triangle Angle Example 1. Triangle Angle Example 2. Triangle Angle Example 3. Challenging Triangle Angle Problem. Proof - Corresponding Angle Equivalence Implies Parallel Lines. Finding more angles. Angles 1. Angles 2. Sum of Interior Angles of a Polygon. Angles of a polygon. Sum of the exterior angles of convex polygon. Introduction to angles (old). Angles (part 2). Angles (part 3). Angles formed between transversals and parallel lines. Angles of parallel lines 2. The Angle Game. Angle Game (part 2). Acute right and obtuse angles. Complementary and supplementary angles. Complementary and supplementary angles. Example using algebra to find measure of complementary angles. Example using algebra to find measure of supplementary angles. Angle addition postulate.

Triangles are not always right (although they are never wrong), but when they are it opens up an exciting world of possibilities. Not only are right triangles cool in their own right (pun intended), they are the basis of very important ideas in analytic geometry (the distance between two points in space) and trigonometry. Pythagorean Theorem. The Pythagorean theorem intro. Pythagorean Theorem 1. Pythagorean Theorem 2. Pythagorean Theorem 3. Pythagorean theorem. Introduction to the Pythagorean Theorem. Pythagorean Theorem II. Garfield's proof of the Pythagorean Theorem. Bhaskara's proof of Pythagorean Theorem. Pythagorean Theorem Proof Using Similarity. Another Pythagorean Theorem Proof. 30-60-90 Triangle Side Ratios Proof. 45-45-90 Triangle Side Ratios. 30-60-90 Triangle Example Problem. Special right triangles. Area of a Regular Hexagon. 45-45-90 Triangles. Intro to 30-60-90 Triangles. 30-60-90 Triangles II. Pythagorean Theorem. The Pythagorean theorem intro. Pythagorean Theorem 1. Pythagorean Theorem 2. Pythagorean Theorem 3. Pythagorean theorem. Introduction to the Pythagorean Theorem. Pythagorean Theorem II. Garfield's proof of the Pythagorean Theorem. Bhaskara's proof of Pythagorean Theorem. Pythagorean Theorem Proof Using Similarity. Another Pythagorean Theorem Proof. 30-60-90 Triangle Side Ratios Proof. 45-45-90 Triangle Side Ratios. 30-60-90 Triangle Example Problem. Special right triangles. Area of a Regular Hexagon. 45-45-90 Triangles. Intro to 30-60-90 Triangles. 30-60-90 Triangles II.

This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.

18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.

Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.

Similar Triangle Basics. Similarity Postulates. Similar triangles 1. Similar Triangle Example Problems. Similar triangles 2. Similarity Example Problems. Solving similar triangles 1. Similarity example where same side plays different roles. Challenging Similarity Problem. Finding Area Using Similarity and Congruence. Similar triangles. Similar triangles (part 2). Similar Triangle Basics. Similarity Postulates. Similar triangles 1. Similar Triangle Example Problems. Similar triangles 2. Similarity Example Problems. Solving similar triangles 1. Similarity example where same side plays different roles. Challenging Similarity Problem. Finding Area Using Similarity and Congruence. Similar triangles. Similar triangles (part 2).

You probably like triangles. You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!. Circumcenter of a Triangle. Circumcenter of a Right Triangle. Three Points Defining a Circle. Area Circumradius Formula Proof. 2003 AIME II Problem 7. Point Line Distance and Angle Bisectors. Incenter and incircles of a triangle. Inradius Perimeter and Area. Angle Bisector Theorem Proof. Angle Bisector Theorem Examples. Angle bisector theorem. Triangle Medians and Centroids. Triangle Medians and Centroids (2D Proof). Medians divide into smaller triangles of equal area. Exploring Medial Triangles. Proving that the Centroid is 2-3rds along the Median. Median Centroid Right Triangle Example. Proof - Triangle Altitudes are Concurrent (Orthocenter). Common Orthocenter and Centroid. Review of Triangle Properties. Euler Line. Euler's Line Proof. Constructing a perpendicular bisector using a compass and straightedge. Constructing a perpendicular line using a compass and straightedge. Constructing an angle bisector using a compass and straightedge. Compass Constructions. Circumcenter of a Triangle. Circumcenter of a Right Triangle. Three Points Defining a Circle. Area Circumradius Formula Proof. 2003 AIME II Problem 7. Point Line Distance and Angle Bisectors. Incenter and incircles of a triangle. Inradius Perimeter and Area. Angle Bisector Theorem Proof. Angle Bisector Theorem Examples. Angle bisector theorem. Triangle Medians and Centroids. Triangle Medians and Centroids (2D Proof). Medians divide into smaller triangles of equal area. Exploring Medial Triangles. Proving that the Centroid is 2-3rds along the Median. Median Centroid Right Triangle Example. Proof - Triangle Altitudes are Concurrent (Orthocenter). Common Orthocenter and Centroid. Review of Triangle Properties. Euler Line. Euler's Line Proof. Constructing a perpendicular bisector using a compass and straightedge. Constructing a perpendicular line using a compass and straightedge. Constructing an angle bisector using a compass and straightedge. Compass Constructions.

GMAT Math: 1. GMAT Math: 2. GMAT Math: 3. GMAT Math: 4. GMAT Math: 5. GMAT: Math 6. GMAT: Math 7. GMAT: Math 8. GMAT: Math 9. GMAT: Math 10. GMAT: Math 11. GMAT: Math 12. GMAT: Math 13. GMAT: Math 14. GMAT: Math 15. GMAT: Math 16. GMAT: Math 17. GMAT: Math 18. GMAT: Math 19. GMAT Math 20. GMAT Math 21. GMAT Math 22. GMAT Math 23. GMAT Math 24. GMAT Math 25. GMAT Math 26. GMAT Math 27. GMAT Math 28. GMAT Math 29. GMAT Math 30. GMAT Math 31. GMAT Math 32. GMAT Math 33. GMAT Math 34. GMAT Math 35. GMAT Math 36. GMAT Math 37. GMAT Math 38. GMAT Math 39. GMAT Math 40. GMAT Math 41. GMAT Math 42. GMAT Math 43. GMAT Math 44. GMAT Math 45. GMAT Math 46. GMAT Math 47. GMAT Math 48. GMAT Math 49. GMAT Math 50. GMAT Math 51. GMAT Math 52. GMAT Math 53. GMAT Math 54.