Spatial reasoning, coordinate geometry, transformations, and trigonometry for geometric problem solving. Covers Euclidean geometry (points, lines, angles, congruence, similarity), triangle geometry (Pythagorean theorem, laws of sines and cosines), circle geometry (tangent lines, inscribed angles, power of a point), coordinate geometry (distance, midpoint, slope, conic sections), transformations as functions (rotation, reflection, translation, dilation), trigonometry (unit circle, identities, inverse functions), and geometric proof techniques. Use when reasoning about shapes, angles, spatial relationships, coordinate problems, or trigonometric computations.
Geometry is the study of shape, size, position, and the properties of space. It grounds abstract mathematics in spatial reasoning and provides the visual intuition that guides discovery across all mathematical domains. This skill covers Euclidean geometry, coordinate methods, transformations, trigonometry, and geometric proof techniques.
Agent affinity: euclid (axiomatic geometry, geometric proofs), euler (coordinate methods, analytic geometry)
Concept IDs: math-shape-properties, math-transformations, math-measurement-area, math-coordinate-geometry, math-trigonometry
Euclid's Elements (c. 300 BCE) builds geometry from five postulates. The first four are uncontroversial; the fifth (the parallel postulate) generated two millennia of investigation and eventually the discovery of non-Euclidean geometries.
Points have no dimension — they represent position only. Lines extend infinitely in both directions and are determined by any two distinct points. Planes are flat, two-dimensional surfaces extending infinitely.
An angle measures the rotation between two rays sharing a common endpoint (vertex).
| Type | Measure | Example |
|---|---|---|
| Acute | 0 < theta < 90 degrees | Interior angles of an equilateral triangle (60 degrees each) |
| Right | theta = 90 degrees | Corner of a rectangle |
| Obtuse | 90 < theta < 180 degrees | Obtuse angle in a 30-30-120 triangle |
| Straight | theta = 180 degrees | A line viewed as an angle |
| Reflex | 180 < theta < 360 degrees | The "outside" of an acute angle |
Key angle relationships:
The angle sum theorem. The interior angles of any triangle sum to 180 degrees. (This is equivalent to the parallel postulate in Euclidean geometry; in hyperbolic geometry the sum is less, in spherical geometry it is greater.)
Triangle classification by sides: Equilateral (all sides equal), isosceles (at least two equal), scalene (all different). By angles: Acute (all angles < 90), right (one angle = 90), obtuse (one angle > 90).
Triangle inequality. For any triangle with sides a, b, c: a + b > c, a + c > b, b + c > a. Three positive numbers form a triangle if and only if each is less than the sum of the other two.
Two figures are congruent if one can be transformed into the other by rigid motions (translations, rotations, reflections) — they have the same shape and size.
Triangle congruence criteria:
Two figures are similar if one can be transformed into the other by a combination of rigid motions and dilation — they have the same shape but possibly different size. The ratio of corresponding sides is the scale factor k.
Triangle similarity criteria:
Worked example. In triangle ABC, DE is parallel to BC with D on AB and E on AC. Prove triangle ADE is similar to triangle ABC.
Proof. Since DE is parallel to BC, angle ADE = angle ABC (corresponding angles with transversal AB) and angle AED = angle ACB (corresponding angles with transversal AC). By AA similarity, triangle ADE ~ triangle ABC.
In a right triangle with legs a, b and hypotenuse c: a^2 + b^2 = c^2.
Proof (Euclid, Elements I.47). Construct squares on each side of the right triangle. The square on the hypotenuse can be dissected into two rectangles, each equal in area to one of the leg squares, established by the congruence of triangles formed by dropping an altitude from the right angle to the hypotenuse.
Converse. If a^2 + b^2 = c^2 for a triangle with sides a, b, c, then the angle opposite c is a right angle.
Generalization. If the angle C opposite side c satisfies c^2 = a^2 + b^2 - 2ab*cos(C) (law of cosines), then the Pythagorean theorem is the special case C = 90 degrees (where cos(90) = 0).
For any triangle with sides a, b, c opposite angles A, B, C:
a/sin(A) = b/sin(B) = c/sin(C) = 2R
where R is the circumradius (radius of the circumscribed circle).
When to use. When you know an angle and its opposite side, plus one other piece (AAS or ASA configurations).
The ambiguous case (SSA). Given angle A, side a, and side b: if a < bsin(A), no triangle exists; if a = bsin(A), exactly one (right) triangle; if b*sin(A) < a < b, two triangles; if a >= b, one triangle.
c^2 = a^2 + b^2 - 2ab*cos(C)
When to use. When you know SAS (two sides and the included angle) or SSS (all three sides, solving for an angle).
Worked example. In a triangle with a = 5, b = 7, C = 60 degrees, find c.
c^2 = 25 + 49 - 2(5)(7)cos(60) = 74 - 70(0.5) = 74 - 35 = 39. So c = sqrt(39) approximately 6.245.
A circle is the set of all points at distance r (the radius) from a center point O. The diameter d = 2r. Circumference = 2pir. Area = pi*r^2.
A tangent to a circle at point P is perpendicular to the radius OP. Two tangent segments from an external point to a circle are equal in length.
Inscribed angle theorem. An inscribed angle (vertex on the circle, sides are chords) is half the central angle subtending the same arc. Consequence: all inscribed angles subtending the same arc are equal.
Thales' theorem (special case). An inscribed angle subtending a diameter is a right angle.
For a point P and a circle, the power of P is d^2 - r^2, where d is the distance from P to the center.
This invariant simplifies many circle geometry problems.
A quadrilateral is cyclic (inscribable in a circle) if and only if opposite angles sum to 180 degrees. Ptolemy's theorem for cyclic quadrilaterals ABCD: AC * BD = AB * CD + AD * BC.
Descartes' coordinate system (Cartesian plane) maps geometric objects to algebraic equations and vice versa, bridging geometry and algebra.
Distance. Between (x_1, y_1) and (x_2, y_2): d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2).
Midpoint. ((x_1 + x_2)/2, (y_1 + y_2)/2).
Slope. m = (y_2 - y_1)/(x_2 - x_1). Undefined for vertical lines.
Parallel lines have equal slopes. Perpendicular lines have slopes whose product is -1 (m_1 * m_2 = -1), provided neither is vertical.
The conic sections — circle, ellipse, parabola, hyperbola — arise from slicing a cone with a plane (Apollonius, c. 200 BCE).
| Conic | Standard form (centered at origin) | Eccentricity |
|---|---|---|
| Circle | x^2 + y^2 = r^2 | e = 0 |
| Ellipse | x^2/a^2 + y^2/b^2 = 1 (a > b > 0) | 0 < e < 1, e = c/a where c^2 = a^2 - b^2 |
| Parabola | y = (1/(4p))x^2 or x = (1/(4p))y^2 | e = 1 |
| Hyperbola | x^2/a^2 - y^2/b^2 = 1 | e > 1, e = c/a where c^2 = a^2 + b^2 |
Worked example. Find the foci of the ellipse x^2/25 + y^2/9 = 1.
Here a^2 = 25, b^2 = 9, so c^2 = 25 - 9 = 16, c = 4. The foci are at (+/-4, 0).
Transformations are functions from the plane to itself. Rigid motions (isometries) preserve distance; similarity transformations preserve shape.
T(x, y) = (x + a, y + b). Every point moves by the same vector (a, b). Preserves distances, angles, and orientation.
R_theta(x, y) = (xcos(theta) - ysin(theta), xsin(theta) + ycos(theta)). Rotates counterclockwise by theta about the origin. Preserves distances and angles.
Across the x-axis: (x, y) -> (x, -y). Across the y-axis: (x, y) -> (-x, y). Across y = x: (x, y) -> (y, x). Preserves distances but reverses orientation.
D_k(x, y) = (kx, ky) for scale factor k > 0. Preserves angles but scales all distances by k. Preserves shape (similarity) but not size (unless k = 1).
Transformations compose: applying R then T gives T(R(x, y)). The composition of two reflections across parallel lines is a translation; the composition of two reflections across intersecting lines is a rotation by twice the angle between the lines.
The unit circle x^2 + y^2 = 1 defines the trigonometric functions: for an angle theta measured counterclockwise from the positive x-axis, the point on the unit circle is (cos(theta), sin(theta)).
Key values:
| theta (degrees) | theta (radians) | sin(theta) | cos(theta) | tan(theta) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| 30 | pi/6 | 1/2 | sqrt(3)/2 | 1/sqrt(3) |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90 | pi/2 | 1 | 0 | undefined |
Pythagorean: sin^2(theta) + cos^2(theta) = 1. Dividing by cos^2 or sin^2 gives: 1 + tan^2(theta) = sec^2(theta), 1 + cot^2(theta) = csc^2(theta).
Angle addition:
Double angle:
Half angle:
arcsin(x) is defined for x in [-1, 1] with range [-pi/2, pi/2]. arccos(x) is defined for x in [-1, 1] with range [0, pi]. arctan(x) is defined for all real x with range (-pi/2, pi/2).
Worked example. Solve 2sin^2(x) - sin(x) - 1 = 0 for x in [0, 2pi).*
Let u = sin(x). Then 2u^2 - u - 1 = 0, so (2u + 1)(u - 1) = 0. Thus u = -1/2 or u = 1.
sin(x) = 1 gives x = pi/2. sin(x) = -1/2 gives x = 7pi/6 and x = 11pi/6.
Solutions: x = pi/2, 7pi/6, 11pi/6.
Synthetic (axiomatic) proofs reason directly from definitions, postulates, and previously proven theorems without coordinates. Euclid's Elements is entirely synthetic.
Strategy: Identify congruent or similar triangles. Most Euclidean geometry proofs reduce to showing two triangles are congruent (via SSS, SAS, ASA, AAS, or HL) or similar (via AA, SAS, or SSS similarity).
Assign coordinates strategically, then compute algebraically.
Worked example. Prove that the diagonals of a parallelogram bisect each other.
Proof. Place the parallelogram with vertices at A = (0, 0), B = (a, 0), C = (a + b, c), D = (b, c). Midpoint of AC: ((a + b)/2, c/2). Midpoint of BD: ((a + b)/2, c/2). The midpoints are identical, so the diagonals bisect each other.
Show that a transformation mapping one figure to another preserves the property in question.
Worked example. Prove that reflections preserve distance.
Proof. The reflection across the x-axis maps (x, y) to (x, -y). The distance between (x_1, y_1) and (x_2, y_2) is sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). The distance between their images (x_1, -y_1) and (x_2, -y_2) is sqrt((x_2 - x_1)^2 + (-y_2 + y_1)^2) = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Equal.
When approaching a geometry problem: