quadratic-lesson

name	quadratic-lesson
description	Use this skill to turn a middle-school quadratic-function comprehensive problem into a compiled interactive lesson page. The agent writes teaching markdown and three declarative JSON specs; repository tools compile the HTML. Parabolas use geometry-spec curves; CSS/JS/runtime are shared with geometry-lesson pages.

Quadratic Lesson

Use this skill when creating or updating a 二次函数综合题 interactive page in the shuxueshuo repository.

The core rule is the same as geometry-lesson: HTML is a compiled artifact. Do not hand-write page HTML, SVG path logic, diagramMarkupFor, drawMini, step navigation, sliders, thumbnails, or page runtime JavaScript.

Output Contract

Work in this order:

If the task is only to iterate this skill, update SKILL.md itself and do not update the knowledge base or case-index.md. The knowledge-base update rules below apply when creating or publishing a lesson page, not when refining the skill instructions.

0. Select knowledge-base references before solving:
   • Read internal/knowledge-points/junior-math-methods.md.
   • Read internal/knowledge-points/case-index.md.
   • Choose one primary pattern, the methods actually needed by the solution, and 1-3 similar published cases.
   • Read the selected cases' 02_solution.md; read their 03_visual_steps.md and lesson-data.json only when the visual flow or JSON structure is directly relevant.
   • If a selected case already demonstrates the JSON shape and id alignment you need, skip the skill's built-in few-shot entirely.
1. Create or update internal/lesson-specs/<problem-id>/01_problem.md.
2. Create or update internal/lesson-specs/<problem-id>/02_solution.md.
3. Create or update internal/lesson-specs/<problem-id>/03_visual_steps.md.
4. Create or update the compiled-page input JSON files:
   • geometry-spec.json (may include curves for y = ax²+bx+c and optional expressionEnv)
   • step-decorations.json
   • lesson-data.json (including meta.classification)
5. Before compiling a publish page, update the knowledge-base metadata:
   • Check whether lesson-data.json.meta.classification is missing, stale, or inconsistent with the final solution; update it before building.
   • If this is a complete JSON-spec page being compiled for publication, add or update the case in internal/knowledge-points/case-index.md.
   • Add one Part 1 row under the chosen pattern, and one Part 2 row under each listed method.
   • Draft pages may include meta.classification, but do not enter case-index.md until the publish compile.
6. Run validation and compilation:

node tools/validate-geometry-spec.mjs internal/lesson-specs/<problem-id>/
node tools/build-lesson-page.mjs internal/lesson-specs/<problem-id>/

The final HTML path is controlled by lesson-data.json.meta.outputPath.

Reference Files

Load only the references needed for the current task:

• Always begin with the knowledge base: ../../knowledge-points/junior-math-methods.md and ../../knowledge-points/case-index.md. Use them to select the primary pattern, allowed middle-school methods, and similar published cases before writing the solution route.
• Always read references/quadratic-solving-principles.md — quadratic-specific modeling rules (coefficient constraints, N-derivation via rotation, expressionEnv ordering, EG+FG optimization pattern, two-curve pattern for Part I vs Part II).
• Always read references/json-schema-guide.md before writing any of the three JSON specs — covers field types, required vs optional, and how to align ids across files.
• Always read references/diagram-drawing-principles.md before writing or revising 03_visual_steps.md or step-decorations.json — covers what to draw, how to mark used values/equalities, constructed segments, moving segments, and geometric transformations.
• Always read references/original-figure-principles.md before adding or changing geometry-spec.originalFigures or any lesson-data.problem.lines[].figures entry. If the source problem has no printed figure, do not invent one.
• Read ../../docs/interactive-lesson-components.md (repo path: internal/docs/interactive-lesson-components.md) before adding or changing sliders, local point controls, draggable-point interactions, custom local grid panels, or step-specific hidden layers. It defines the relationship between the main parameter slider, step-local point controls, and hideLayers / grid.panels.
• Read references/nankai-25-fewshot.md only as a fallback when case-index.md has no sufficiently similar published case, or when the selected cases do not demonstrate the JSON shape you need. Do not prefer this fixed few-shot over a closer indexed case.
• Read the real schema files before writing JSON (they override anything in the reference docs if there is a conflict):
  • internal/schemas/geometry-spec.schema.json (expressionEnv, curves, optional basePolygon/movingPolygon)
  • internal/schemas/step-decorations.schema.json (parabola, axisOfSymmetry, vertex, curvePoint, dashedLine)
  • internal/schemas/lesson-data.schema.json

Do not use skill references as JS/CSS sources.
Do not read geometry-solving-principles.md or nankai-24-fewshot.md for quadratic problems — those are geometry-specific (polygon clipping, rotation angles, area decompositions).

Style Preset and Normalizer

The build pipeline runs a normalizer (tools/lib/lesson-normalizer.mjs) before compilation and validation. It reads internal/config/style-presets.json and fills in any missing style fields on decoration elements by type. It also auto-generates range: [t, t] for non-movable steps that lack a range.

What this means for writing JSON specs:

• Omit default style fields such as color, width, dash, r, size, fontSize when the preset value is acceptable. The normalizer will fill them in.
• Only declare a style field when you need a value different from the preset — for example, a lighter parabola for background context, or a custom point radius.
• Always declare semantic fields that the normalizer does not handle when you use them: decoration fields such as at, from, to, label, labelText, text, curveId, xExpr, vertex, rayA, rayB, dx, dy, showLabel, offsetPx, labelRadius, lockLabel; step-level fields such as domain, pointOverrides, hideLayers; and grid-element fields such as panels.
• Always declare range for movable steps (movable: true). Non-movable steps may omit range.

See internal/config/style-presets.json for the full list of types and their defaults.

Audience — quadratic-specific guidance

Default to middle-school students. Compared with pure folding/rotation geometry pages:

• Balance 代数推导（解析式、配方、判别式、根与系数）with 坐标图示（抛物线、对称轴、与坐标轴交点）。
• Many steps justify coefficients before drawing conclusions about symmetric axis / vertices / intersections with axes.
• Do not create a separate "公共结论" section when the reusable fact naturally arises inside a sub-question. Extract the fact at the exact solving step where it is first needed, then cite or reuse it in the next step.
• For middle-school trigonometry, use only the definition of tan in a right triangle. Do not use tangent subtraction/addition formulas, slope-angle tangent formulas, or other high-school trigonometric identities. If an angle condition involves (45^\circ), prefer constructing an auxiliary point/line that creates a right triangle where the target tangent can be read as opposite leg divided by adjacent leg.
• Follow references/diagram-drawing-principles.md for step diagrams, including how to mark used quantities, equalities, constructed segments, moving segments, and geometric transformations.
• If part II introduces constructions (直角、等腰、中点、线段比例), treat those steps like geometry: draw only what the derivation truly uses.
• Dynamic controls usually correspond to a parameter letter shown on the exam (m, t, etc.). Align geometry-spec.movingParam, expressions that reference this letter, lesson-data.ui.sliderLabel, and lesson-data.policies[].range.
• Do not add a main slider for a coefficient or unknown that the problem asks students to solve (a、b、c). Keep those steps non-movable and use a representative symbolic/solved drawing state only for the diagram. Use local controls only for genuine moving points inside a construction or shortest-path observation.
• Keep numeric endpoints consistent everywhere (≤, <) across markdown, JSON chips, and slider ranges.
• Prefer the student's natural solving order over a compact formula-first route. If coefficients are already numeric, substitute them first and then complete the square; do not start with a general vertex formula unless the symbolic form is itself the point of the step.
• Keep each step's algebra scoped to the current idea. Do not solve parameter relations such as a-r-c early merely because they are available; save them for the step where the problem actually asks to determine the coefficient.

Step 1: 01_problem.md

Same structural headings as geometry-lesson (基本信息、题目原文、已知条件、小问列表、标准答案). Highlight:

• the quadratic relation (y = ax^2 + bx + c) and constraints among coefficients (e.g. (2a + b = 0)).
• axis of symmetry wording (“抛物线的对称轴与 x 轴交于点 D”).
• moving-point wording (“点 E、G …”，“线段”“比例”“最小值”).

Step 2: 02_solution.md

Use short derivation lines with ∵ / ∴. Prefer titles such as:

• 由已知系数关系消参
• 由对称轴求交点坐标
• 配方或代入定点写解析式
• 化简函数表达式并求点坐标
• 构造直角三角形求线段
• 构造等腰直角三角形转化线段
• 将军饮马，折线三点共线时最短

Avoid dumping lengthy algebraic manipulation without naming intermediate meanings each step.

Student-facing sequencing rules:

• When a sub-question gives numeric coefficients, first write the concrete function and then derive the requested object (y=-2x²+8x+6 → y=-2(x-2)²+14 → D(2,14)).
• Every key coordinate must name its source: which two lines are intersected, which foot is constructed, which fixed length is read, or which special triangle supplies the coordinate.
• If a distance condition gives two candidate positions, list both and then use the quadrant, opening direction, or point-on-curve condition to choose.
• For an auxiliary foot that drives an area or length computation, give the foot a visible point name and use the same name in the derivation (MG, not a hidden h) so the formula maps directly to the diagram.
• In a multi-step path-minimum argument, first record only the facts needed for the conversion (for example A(r,0), AA′=8-r, and P(r/2,√3r/2)), then do the path conversion, then use the final minimum to solve the coefficient.

When a coordinate-geometry condition contains 90° or equal lengths, prefer a middle-school geometry route before coefficient solving: draw the needed perpendicular foot, prove right-triangle congruence or an isosceles-right relation, obtain the target point coordinate, then substitute into the parabola.

When an angle condition determines a line through an axis point, first look for a middle-school construction before using slope language. Useful patterns include reflecting a point across an axis, building an isosceles triangle, or letting the target line meet an axis at an auxiliary point (C') so equal angles become equal vertical segments such as (C'O=CO). Use slope only after the construction has produced two points on the line, and phrase it as "由两点确定直线" when possible.

When computing a coordinate triangle area, first check whether a vertical or horizontal auxiliary segment can split the triangle into two readable pieces. Prefer an "铅垂面积" expression such as 1/2·vertical base·left horizontal distance + 1/2·vertical base·right horizontal distance over determinant-style coordinate area in student-facing text.

For geometry-heavy quadratic综合题, prefer a visible geometric derivation before coordinate formulas:

• When a square or rotated side determines a point such as G, avoid saying only "rotate 90° to get coordinates". First draw the needed perpendicular foot, prove the right-triangle congruence, and then read the horizontal and vertical distances.
• Every coordinate expression like G(x(t), y0) must explain the source of each coordinate separately: the horizontal part from an equal segment or distance, and the vertical part from a fixed line, foot, or trajectory.
• When a parallelogram determines a moving point such as G, do not present a vector formula in student-facing text. Read the same horizontal and vertical changes from one side of the parallelogram and apply them to the adjacent vertex, then state the trajectory of G.
• Avoid introducing a new auxiliary variable merely to shorten arithmetic when the original expressions are readable. Keep the problem's own parameters visible unless the substitution reveals a real structure.
• If a conclusion uses a midpoint or a shortest-position collinearity, state why the chosen point is the midpoint or intersection before using its coordinate.
• Reuse prior conclusions through reference components in lesson-data.json instead of repeating a full derivation block.
• For nested shortest-path arguments, split by idea rather than by algebra length. A typical sequence is: fixed moving-point state and reflection straightening; auxiliary angle proof; moving the remaining point and applying perpendicular-distance shortest; final computation.
• If a path-minimum conclusion can be reduced to one decisive segment such as √2·DG, compute from that segment directly. Do not derive unused optimal-point coordinates or side lengths merely because they are available.
• For two-moving-point distance sums and shortest-path coordinate computation, use references/quadratic-solving-principles.md §Double-Moving Point To Single-Moving Point Path.
• For weighted or double-moving path sums, make the conversion step explicit in the step title and derivation: construct the parallelogram/translation that turns one moving segment into a segment from a fixed or linked point, construct the special right triangle that absorbs the weight, and state that the two-moving-point problem has become a one-moving-point broken path.
• For a path such as BF+BG, first construct a parallelogram to replace BG by a segment ending at the same moving point, then use symmetry/reflection to replace the other segment if available, and only then apply the broken-line or 将军饮马 inequality.
• Mark fixed distances as fixed before optimizing. For example, when EF is the distance between two parallel lines, state that it is independent of the moving point before rewriting the target expression.
• Do not over-explain coordinates inside the optimization step if a prior step already established them. Move reusable point coordinates such as P(r/2,√3r/2) into an earlier setup step, then cite them compactly during the shortest-path computation.

Step 3: 03_visual_steps.md

Same layering mindset as geometry-lesson (whole-problem / section / phase / step). For quadratic pages additionally specify:

• which subset of the curve domain should remain visible for readability (geometry-spec.domain).
• when expressionEnv should expose coefficients (a, b, c) separately so fixed-point formulas (["0","c"]) stay declarative.
• when to use decorations parabola, axisOfSymmetry, vertex, curvePoint versus ordinary point, following references/diagram-drawing-principles.md.
• which labels are allowed at each stage: use symbolic coordinates before a parameter is solved; reserve final numeric coordinates and final answer labels for the step that derives them.
• where local zoom domains are needed. Geometry-heavy congruence or shortest-path computation steps should often use steps[stepId].domain so the construction, angle marks, and used lengths are readable.
• which diagram labels are new visual information. Do not add formula cards or labels that merely repeat the derivation panel or the step box.
• which objects should disappear after their job is done. In a final coefficient-solving step, remove path-construction points such as E,F,N,H if the calculation only needs A,C,P and the parabola.
• when a step compares multiple solved coordinate states. If each state has a different meaningful origin/y-axis, follow references/diagram-drawing-principles.md §Quadratic-Specific Drawing and references/json-schema-guide.md §Grid panels and hidden layers.
• how stepLabels name each step. Prefer compact "method + target" labels such as 等角作C′定BM, 铅垂面积求b, or 构造等腰求a instead of vague result labels such as 确定 BM or 求 b.
• For path-minimum navigation, use method-forward labels such as 双动点转单动点, 将军饮马求最小值, and 代入最小值求a.

Use review-friendly visual discipline:

• Keep SVG diagram labels symbolic/ASCII where possible. See references/diagram-drawing-principles.md §Step Relevance / §Mark Used Quantities.
• Fill congruent triangles or corresponding construction triangles with the same light color so students can see the matched shapes before reading the text.
• Keep labels limited to the quantities used in the current derivation. For a line-segment conversion step, prefer the two decisive labels such as FM=1/2AE and FH=1/2AG; omit midpoint labels if the derivation panel already states them.
• For dense shortest-path computation diagrams and label/box occlusion checks, use references/diagram-drawing-principles.md §Mark Used Quantities and §Step Relevance.
• Local zoom should focus on the active construction while preserving necessary context such as the moving point trajectory. Do not leave large empty regions when a tighter domain would make the construction clearer.
• Split steps when two different ideas are being taught, such as "derive G's trajectory" and "apply reflection shortest path". Each idea may have its own local control if dragging helps the student see it.
• For a fixed-point reflection step, use local controls for the points that are genuinely moving in that fixed state. For a later step where the fixed point itself moves, switch the local control to that point and keep dependent points linked or hidden if they are no longer the focus.
• For linked moving points, a dual local-control display is useful only when it makes the dependency visible. Label the controlling point and the linked point consistently, and state in the derivation that the second point moves with the first rather than being an independent variable.
• When a coordinate formula is derived from a distance, show the corresponding auxiliary segment or projection in the diagram, for example a horizontal distance from A to the projection of G.
• Keep navigation labels synchronized with step titles and make them method-based, e.g. 中线中位线转线段, 推导G轨迹, 将军饮马求最小值.
• If a step computes a perpendicular distance without using a distance formula, draw the small right-triangle decomposition used by middle-school geometry. Add auxiliary points such as J,K only when they make the length computation visible; remove whole-length labels that duplicate the derivation when component lengths are clearer.

Step 4: JSON Specs

geometry-spec.json

• Always define domain, fixedPoints, movingPoints, movingParam.
• Optional basePolygon / movingPolygon for folding/overlap-style problems (omit entirely on pure coordinate-parabola pages).
• Use expressionEnv as an ordered list { "name": "c", "expr": "-5" } so downstream expressions can reference coefficients symbolically.
• Declare parabolas:

"curves": [
  { "id": "parabolaMain", "type": "parabola", "a": "a", "b": "b", "c": "c" }
]

(a/b/c may also live under "params" — both shapes are supported.)

• Intersections still use derivedIntersections whenever possible (same rule as geometry specs).

step-decorations.json

Besides geometry decorators, you may use:

type	purpose
parabola	draws sampled curve from state.curves[curveId]
axisOfSymmetry	vertical line (x = -b/(2a))
vertex	vertex marker from coefficients
curvePoint	(xExpr, y) from curve via xExpr evaluated with state.env

For separated coordinate cases, use grid.panels together with steps[stepId].hideLayers; details live in references/json-schema-guide.md §Grid panels and hidden layers.

lesson-data.json

Same constraints as geometry-lesson: no HTML strings; legend rows remain declarative; step ids must align across steps, policies, stepLabels, and step-decorations.

Hard constraints:

• meta.classification.pattern must be one primary pattern ID defined in junior-math-methods.md.
• meta.classification.methods must list only method IDs from junior-math-methods.md, ordered by first use in the solution.
• Classification must match both the final 02_solution.md and any case-index.md rows added during publish compilation.

Validation And Compilation

Before compiling a page for publication:

• Re-read lesson-data.json.meta.classification and confirm the pattern / methods still match the final solution.
• Re-read internal/knowledge-points/case-index.md; if the current complete JSON-spec page is being published and is missing from the index, add or update its Part 1 and Part 2 rows before building.
• Do not add draft or unreviewed pages to case-index.md.

Always run:

node tools/validate-geometry-spec.mjs internal/lesson-specs/<problem-id>/
node tools/build-lesson-page.mjs internal/lesson-specs/<problem-id>/

Then spot-check the HTML locally.

Final Review Checklist

• Problem metadata matches the exam reference (problem-id, title).
• expressionEnv order reflects algebraic dependency (b after a, etc.).
• Slider labels explain what moves (sliderLabel, paramLabelPrefix).
• No main slider is used for coefficients that are being solved.
• Every lesson step has diagram intent documented in 03_visual_steps.md.
• Earlier diagrams do not contain later conclusions: no solved coordinates, final coefficient values, helper points, or final curve state before the matching derivation step.
• Setup steps do not prematurely solve parameters that are only needed after the minimum or existence condition is known.
• Known roots/intercepts are used to simplify/factor the parabola directly before introducing any new unknown point parameter.
• Geometry conditions are solved geometrically when possible: perpendicular feet, right-triangle congruence, isosceles-right triangles, and 将军饮马 before coordinate/vector formulas.
• Coordinate claims for constructed points explain both horizontal and vertical origins, and the diagram marks the line or distance used to read them.
• Key points from line intersections, such as Q, explicitly state the two objects being intersected before the coordinate is written.
• Auxiliary feet used in formulas are named consistently in text and diagram, and the formula uses that name.
• For two-moving-point path expressions, natural construction language, and shortest-path coordinate computation discipline, check references/quadratic-solving-principles.md §Double-Moving Point To Single-Moving Point Path.
• Angle conditions are converted with visible auxiliary geometry when possible: symmetry points, isosceles triangles, or axis-intersection points before slope/tangent formulas.
• Coordinate-area steps use vertical/horizontal split areas when the diagram provides a natural base, before determinant formulas.
• Diagram labels are not duplicating the derivation panel: no repeated formula cards when the same result is already in derive or box.
• Final computation diagrams remove stale shortest-path construction objects and show only the objects needed for the current substitution.
• SVG diagrams contain no Chinese explanatory text; dense diagrams keep decisive labels only; conclusion boxes do not cover important points or paths. See references/diagram-drawing-principles.md.
• Multiple coordinate states with different origins/axes use hideLayers + grid.panels instead of a misleading continuous coordinate plane. See references/json-schema-guide.md.
• Step navigation labels are short but meaningful, usually "method + target".
• Congruent or corresponding triangles that drive a step use matching light fills, and nonessential labels are removed from dense diagrams.
• Long derivations are split when a trajectory result, a segment conversion, and an optimization argument are separate ideas.
• For reflection/path-minimum steps, the filled region should be the triangle or quadrilateral that drives the current conclusion, not the final answer triangle by habit. If the step proves △G₁DG₂ is right-isosceles, fill △G₁DG₂; if it proves ∠QDM=45°, fill △DMQ.
• Final length computation should use the shortest established expression directly. If the previous step gives 最小周长=√2·DG and DG=3m, solve from 3√2m=9√2 instead of recomputing EF+FG+GE.
• JSON contains zero HTML fragments.
• Step ids stay synchronized across all artifacts.
• lesson-data.meta.classification.pattern and every listed method ID exist in junior-math-methods.md.
• Similar cases from case-index.md were considered before using a fixed few-shot.
• For publish compilation, case-index.md contains the current page under its pattern and every listed method.
• Validation + compilation succeed without patching generated HTML.