🗺️ Geometry Concept Atlas

Quick primers for each major geometry topic, showing how concepts connect and appear in AMC problems.

🔺 Triangle Fundamentals

Basic Triangles

Core Concept: Triangles are the building blocks of plane geometry, defined by three non-collinear points. The study of triangles encompasses congruence (same shape and size), similarity (same shape, different size), and special properties of isosceles and equilateral triangles. AMC Appearance: Problems often test recognition of congruent triangles (SSS, SAS, ASA, AAS) and similar triangles, especially in multi-step proofs or when finding unknown lengths.

Triangle Centers

Core Concept: Four special points exist in every triangle: the centroid (intersection of medians), incenter (intersection of angle bisectors), circumcenter (intersection of perpendicular bisectors), and orthocenter (intersection of altitudes). These centers have unique properties and relationships, including the famous Euler line connecting three of them. AMC Appearance: Problems frequently ask for distances between centers, properties of the Euler line, or use these centers as stepping stones in larger constructions.

Special Segments

Core Concept: Medians, altitudes, and angle bisectors are the three fundamental cevians in triangles, each with unique properties. Medians divide the triangle into six equal areas, altitudes create right angles, and angle bisectors satisfy the Angle Bisector Theorem. AMC Appearance: These segments often appear in ratio problems, area calculations, or as tools for proving other geometric relationships.

📐 Angle Relationships

Angle Chasing

Core Concept: The art of finding unknown angles through systematic application of angle relationships: vertical angles, linear pairs, parallel line theorems, triangle angle sums, and exterior angle theorems. Success requires recognizing patterns and building chains of equal angles. AMC Appearance: Pure angle problems are common, especially involving parallel lines, triangles, or circles. The key is identifying the right starting point and following the chain.

Parallel Lines

Core Concept: When two lines are parallel, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary. These relationships create similar triangles and enable many geometric proofs. AMC Appearance: Parallel lines often appear in trapezoids, parallelograms, or as auxiliary lines in complex figures. They’re a common source of similar triangles.

🔄 Similarity and Ratios

Similarity Theory

Core Concept: Two figures are similar if corresponding angles are equal and corresponding sides are proportional. Similarity preserves angle measures while scaling lengths by a constant factor. This creates powerful tools for finding unknown lengths and areas. AMC Appearance: Similar triangles appear in countless problems, often disguised as parallel lines, angle bisectors, or circle properties. The key is recognizing the similarity and setting up the correct ratio.

Angle Bisector Theorem

Core Concept: An angle bisector divides the opposite side into segments proportional to the adjacent sides. This creates a powerful tool for finding unknown lengths when angle bisectors are present. AMC Appearance: Problems often give angle bisector information and ask for side lengths or ratios. The theorem provides a direct path to the answer.

⭕ Circle Properties

Circles and Power of a Point

Core Concept: Circles have fundamental properties involving chords, tangents, and secants. The Power of a Point theorem unifies these relationships: for any point P, the product of distances to two intersection points of a line through P with the circle is constant. AMC Appearance: Circle problems frequently test chord lengths, tangent properties, and Power of a Point applications. The key is identifying which relationship applies.

Cyclic Quadrilaterals

Core Concept: A quadrilateral is cyclic if all four vertices lie on a circle. This happens if and only if opposite angles are supplementary. Cyclic quadrilaterals have special properties, including Ptolemy’s theorem for side lengths. AMC Appearance: Problems often test recognition of cyclic quadrilaterals and application of their properties. Look for equal subtended angles or supplementary opposite angles.

📏 Length and Area

Classical Formulas

Core Concept: Heron’s formula finds triangle area from side lengths, while Brahmagupta’s formula does the same for cyclic quadrilaterals. Stewart’s theorem relates cevian lengths to side lengths, and Apollonius’s theorem is a special case for medians. AMC Appearance: These formulas appear in problems with specific side lengths or when other methods become too complex. They’re often the most direct path to the answer.

Area Ratios

Core Concept: When triangles share heights or bases, their areas are proportional to their bases or heights respectively. This creates powerful tools for finding area ratios without calculating individual areas. AMC Appearance: Problems often present complex figures where area ratios can be found through shared heights or bases, avoiding lengthy calculations.

🔄 Transformations

Basic Transformations

Core Concept: Reflections, rotations, translations, and homothety (scaling) preserve certain geometric properties while changing others. These transformations can simplify problems by moving figures to more convenient positions. AMC Appearance: Transformation problems often involve reflecting points across lines, rotating figures, or using homothety to create similar figures. The key is choosing the right transformation.

Coordinate Geometry

Core Concept: Placing geometric figures in coordinate systems allows algebraic methods to solve geometric problems. The shoelace formula finds areas, distance formula finds lengths, and circle equations describe circular relationships. AMC Appearance: When pure geometry becomes complex, coordinate methods often provide a direct path to the answer. Problems may explicitly give coordinates or require setting up a coordinate system.

🎲 Probability and 3D

Geometric Probability

Core Concept: Probability problems where the sample space is geometric (lengths, areas, or volumes). Success requires finding the ratio of favorable outcomes to total outcomes using geometric calculations. AMC Appearance: Problems often involve random points on line segments, random chords in circles, or random points in regions. The key is setting up the correct geometric ratio.

3D Geometry

Core Concept: Extending 2D concepts to three dimensions, including volumes, surface areas, and projections. Many 3D problems can be solved by considering 2D cross-sections or projections. AMC Appearance: 3D problems often involve cubes, prisms, or spheres. The key is identifying the relevant 2D cross-section or using projection methods.

🔗 Advanced Techniques

Mass Points

Core Concept: Assigning masses to points in a triangle to find ratios using the principle of moments. This provides an alternative to coordinate geometry for ratio problems. AMC Appearance: Problems involving cevians, medians, or angle bisectors often yield to mass point methods. The key is choosing appropriate masses to simplify calculations.

Ceva and Menelaus

Core Concept: Ceva’s theorem gives conditions for three cevians to be concurrent, while Menelaus’s theorem gives conditions for three points to be collinear. Both involve products of ratios. AMC Appearance: These theorems appear in problems about concurrency or collinearity. The key is identifying the right configuration and setting up the product of ratios.

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