Press n or j to go to the next uncovered block, b, p or k for the previous block.
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import { BYRNE } from '../types'
function getProp4Tutorial(isTouch: boolean): TutorialSubStep[][] {
const tapHold = isTouch ? 'Tap and hold' : 'Click and hold'
return [
// ── Step 0: Draw segment EF (straightedge) ──
[
{
instruction: `${tapHold} point {pt:E}`,
speech: isTouch
? "We're given two triangles with two sides and the included angle equal. Complete triangle DEF by pressing and holding on point E."
: "We're given two triangles with two sides and the included angle equal. Complete triangle DEF by clicking and holding on point E.",
hint: { type: 'point', pointId: 'pt-E' },
advanceOn: null,
},
],
// ── Step 1: Superposition (drag triangle ABC onto DEF) ──
[
{
instruction: 'Drag triangle ABC onto triangle DEF',
speech: 'Now let us see if the triangles coincide. Drag the triangle across to the other.',
hint: { type: 'triangle-highlight', ids: ['pt-A', 'pt-B', 'pt-C'] },
advanceOn: null,
},
],
]
}
/**
* Proposition I.4 — SAS Congruence (Theorem)
*
* If two triangles have two sides equal to two sides respectively,
* and have the angles contained by the equal straight lines equal,
* then they also have the base equal to the base, the triangle equals
* the triangle, and the remaining angles equal the remaining angles.
*
* Setup: Two triangles ABC (left) and DEF (right).
* Given: AB = DE, AC = DF, ∠BAC = ∠EDF
* Triangle DEF is missing segment EF.
* The user draws EF, and C.N.4 (superposition) derives BC = EF.
*
* Coordinates are constructed so that the equalities hold exactly:
* DEF is formed by rotating the vectors (B−A) and (C−A) by angle θ
* and translating to D.
*/
// ── Default positions ──
const DEFAULT_A = { x: -4, y: -0.5 }
const DEFAULT_B = { x: -6.2, y: -1.5 }
const DEFAULT_C = { x: -2.8, y: 1.8 }
const DEFAULT_D = { x: 2.5, y: -0.5 }
const THETA = 0.4 // rotation angle in radians
/**
* Recompute all given elements from current draggable point positions.
* E and F are derived from A, B, C, D to maintain SAS invariants:
* E = D + Rot(θ)·(B−A), F = D + Rot(θ)·(C−A)
*/
export function computeProp4GivenElements(
positions: Map<string, { x: number; y: number }>
): ConstructionElement[] {
const A = positions.get('pt-A') ?? DEFAULT_A
const B = positions.get('pt-B') ?? DEFAULT_B
const C = positions.get('pt-C') ?? DEFAULT_C
const D = positions.get('pt-D') ?? DEFAULT_D
const cosT = Math.cos(THETA)
const sinT = Math.sin(THETA)
const E = {
x: D.x + cosT * (B.x - A.x) - sinT * (B.y - A.y),
y: D.y + sinT * (B.x - A.x) + cosT * (B.y - A.y),
}
const F = {
x: D.x + cosT * (C.x - A.x) - sinT * (C.y - A.y),
y: D.y + sinT * (C.x - A.x) + cosT * (C.y - A.y),
}
return [
// Triangle ABC — all 3 points + 3 segments
{ kind: 'point', id: 'pt-A', x: A.x, y: A.y, label: 'A', color: BYRNE.given, origin: 'given' },
{ kind: 'point', id: 'pt-B', x: B.x, y: B.y, label: 'B', color: BYRNE.given, origin: 'given' },
{ kind: 'point', id: 'pt-C', x: C.x, y: C.y, label: 'C', color: BYRNE.given, origin: 'given' },
{
kind: 'segment',
id: 'seg-AB',
fromId: 'pt-A',
toId: 'pt-B',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-AC',
fromId: 'pt-A',
toId: 'pt-C',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-BC',
fromId: 'pt-B',
toId: 'pt-C',
color: BYRNE.given,
origin: 'given',
},
// Triangle DEF — 3 points + 2 segments (EF missing — user draws it)
{ kind: 'point', id: 'pt-D', x: D.x, y: D.y, label: 'D', color: BYRNE.given, origin: 'given' },
{ kind: 'point', id: 'pt-E', x: E.x, y: E.y, label: 'E', color: BYRNE.given, origin: 'given' },
{ kind: 'point', id: 'pt-F', x: F.x, y: F.y, label: 'F', color: BYRNE.given, origin: 'given' },
{
kind: 'segment',
id: 'seg-DE',
fromId: 'pt-D',
toId: 'pt-E',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-DF',
fromId: 'pt-D',
toId: 'pt-F',
color: BYRNE.given,
origin: 'given',
},
] as ConstructionElement[]
}
// ── Compute initial positions ──
const cosT = Math.cos(THETA)
const sinT = Math.sin(THETA)
const baDx = DEFAULT_B.x - DEFAULT_A.x
const baDy = DEFAULT_B.y - DEFAULT_A.y
const E = {
x: DEFAULT_D.x + cosT * baDx - sinT * baDy,
y: DEFAULT_D.y + sinT * baDx + cosT * baDy,
}
const caDx = DEFAULT_C.x - DEFAULT_A.x
const caDy = DEFAULT_C.y - DEFAULT_A.y
const F = {
x: DEFAULT_D.x + cosT * caDx - sinT * caDy,
y: DEFAULT_D.y + sinT * caDx + cosT * caDy,
}
export const PROP_4: PropositionDef = {
id: 4,
title:
'If two triangles have two sides and the included angle equal, the triangles are congruent',
kind: 'theorem',
resultSegments: [
{ fromId: 'pt-B', toId: 'pt-C' },
{ fromId: 'pt-E', toId: 'pt-F' },
],
givenFacts: [
{
left: { a: 'pt-A', b: 'pt-B' },
right: { a: 'pt-D', b: 'pt-E' },
statement: 'AB = DE',
},
{
left: { a: 'pt-A', b: 'pt-C' },
right: { a: 'pt-D', b: 'pt-F' },
statement: 'AC = DF',
},
],
givenAngles: [
{ spec: { vertex: 'pt-A', ray1End: 'pt-B', ray2End: 'pt-C' }, color: BYRNE.red },
{ spec: { vertex: 'pt-D', ray1End: 'pt-E', ray2End: 'pt-F' }, color: BYRNE.red },
],
equalAngles: [
[
{ vertex: 'pt-A', ray1End: 'pt-B', ray2End: 'pt-C' },
{ vertex: 'pt-D', ray1End: 'pt-E', ray2End: 'pt-F' },
],
],
givenAngleFacts: [
{
left: { vertex: 'pt-A', ray1: 'pt-B', ray2: 'pt-C' },
right: { vertex: 'pt-D', ray1: 'pt-E', ray2: 'pt-F' },
statement: '∠BAC = ∠EDF',
},
],
theoremConclusion: '△ABC = △DEF\n∠ABC = ∠DEF, ∠ACB = ∠DFE',
superpositionFlash: {
pairs: [
{ src: 'pt-A', tgt: 'pt-D' },
{ src: 'pt-B', tgt: 'pt-E' },
{ src: 'pt-C', tgt: 'pt-F' },
],
triA: ['pt-A', 'pt-B', 'pt-C'],
triB: ['pt-D', 'pt-E', 'pt-F'],
},
draggablePointIds: ['pt-A', 'pt-B', 'pt-C', 'pt-D'],
computeGivenElements: computeProp4GivenElements,
givenElements: [
// Triangle ABC — all 3 points + 3 segments
{
kind: 'point',
id: 'pt-A',
x: DEFAULT_A.x,
y: DEFAULT_A.y,
label: 'A',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'point',
id: 'pt-B',
x: DEFAULT_B.x,
y: DEFAULT_B.y,
label: 'B',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'point',
id: 'pt-C',
x: DEFAULT_C.x,
y: DEFAULT_C.y,
label: 'C',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-AB',
fromId: 'pt-A',
toId: 'pt-B',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-AC',
fromId: 'pt-A',
toId: 'pt-C',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-BC',
fromId: 'pt-B',
toId: 'pt-C',
color: BYRNE.given,
origin: 'given',
},
// Triangle DEF — 3 points + 2 segments (EF missing — user draws it)
{
kind: 'point',
id: 'pt-D',
x: DEFAULT_D.x,
y: DEFAULT_D.y,
label: 'D',
color: BYRNE.given,
origin: 'given',
},
{ kind: 'point', id: 'pt-E', x: E.x, y: E.y, label: 'E', color: BYRNE.given, origin: 'given' },
{ kind: 'point', id: 'pt-F', x: F.x, y: F.y, label: 'F', color: BYRNE.given, origin: 'given' },
{
kind: 'segment',
id: 'seg-DE',
fromId: 'pt-D',
toId: 'pt-E',
color: BYRNE.given,
origin: 'given',
},
{
kind: 'segment',
id: 'seg-DF',
fromId: 'pt-D',
toId: 'pt-F',
color: BYRNE.given,
origin: 'given',
},
] as ConstructionElement[],
steps: [
{
instruction: 'Join {pt:E} to {pt:F}',
expected: { type: 'straightedge', fromId: 'pt-E', toId: 'pt-F' },
highlightIds: ['pt-E', 'pt-F'],
tool: 'straightedge',
citation: 'Post.1',
},
{
instruction: 'Place triangle {tri:ABC} onto triangle {tri:DEF}',
expected: {
type: 'superposition',
src: ['pt-A', 'pt-B', 'pt-C'],
tgt: ['pt-D', 'pt-E', 'pt-F'],
mapping: [
['pt-A', 'pt-D'],
['pt-B', 'pt-E'],
['pt-C', 'pt-F'],
],
establishes: {
congruence: { statement: '△ABC ≅ △DEF' },
cascade: [
{
kind: 'segment-equality',
params: { leftA: 'pt-B', leftB: 'pt-C', rightA: 'pt-E', rightB: 'pt-F' },
statement: 'BC = EF',
justification: 'C.N.4: Triangles coincide by superposition',
},
{
kind: 'angle-equality',
params: {
leftVertex: 'pt-B',
leftRay1: 'pt-A',
leftRay2: 'pt-C',
rightVertex: 'pt-E',
rightRay1: 'pt-D',
rightRay2: 'pt-F',
},
statement: '∠ABC = ∠DEF',
justification: 'C.N.4: Remaining angles coincide',
},
{
kind: 'angle-equality',
params: {
leftVertex: 'pt-C',
leftRay1: 'pt-A',
leftRay2: 'pt-B',
rightVertex: 'pt-F',
rightRay1: 'pt-D',
rightRay2: 'pt-E',
},
statement: '∠ACB = ∠DFE',
justification: 'C.N.4: Remaining angles coincide',
},
],
},
},
highlightIds: ['pt-A', 'pt-B', 'pt-C'],
tool: null,
citation: 'C.N.4',
},
],
getTutorial: getProp4Tutorial,
explorationNarration: {
introSpeech:
'You proved two triangles are congruent! Now drag the points to test it with different triangles.',
pointTips: [
{
pointId: 'pt-A',
speech:
'See how triangle DEF reshapes to keep matching? The congruence holds for any vertex position.',
},
{
pointId: 'pt-B',
speech: 'Watch the matching side of DEF change too. Both triangles stay congruent!',
},
{
pointId: 'pt-C',
speech: 'See how EF always equals BC? The third side always matches.',
},
{
pointId: 'pt-D',
speech:
"Watch the congruence hold everywhere. It doesn't matter where the second triangle sits!",
},
],
},
}
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