Three vertices, two arrows, one shadow · all of it held by hand
01
The triangle
Grab any corner. The angles rearrange — the sum never moves.
Three points, three sides, three angles — and one iron law: the angles always share exactly $180°$ between them. Drag a vertex anywhere; watch one angle steal from the others and never break the bank. Everything else — sides by the law of cosines, the area, the triangle’s very name — recomputes as you pull.
The corners
angle A—
angle B—
angle C—
their sum—
area—
its name—
drag any vertex — the sum of angles is the one thing you cannot change
02
Pythagoras
Not a formula — a fact about real squares you can watch balance.
Build actual squares on the three sides of a right triangle. The theorem says the two gold ones, together, weigh exactly as much as the camellia one: $a^2+b^2=c^2$. Stretch the legs and watch the balance hold to the last decimal — it is an equation of surfaces, not of letters.
The legs
a3
b4
a²—
b²—
a²+b²—
c²—
c—
the balance—
gold squares on the legs · camellia square on the hypotenuse · same total, always
Right angles onlytilt the corner and $a^2+b^2$ stops being $c^2$ — the general law is $c^2=a^2+b^2-2ab\cos C$, and Pythagore is its $90°$ special case.
03
The vectors
An arrow is not a place — it is a displacement, and displacements add.
Drag the two arrowheads. Their sum $\vec u+\vec v$ is the diagonal of the parallelogram they span: go along $\vec u$, then along $\vec v$, and the emerald arrow is where you ended up. Coordinates just do this addition one axis at a time.
The arrows
u—
v—
u + v—
‖u + v‖—
gold: u · marine: v · emeraude: their sum, the diagonal
04
The dot product
One number that knows the angle: positive ahead, zero across, negative behind.
$\vec u\cdot\vec v = u_xv_x+u_yv_y = \Vert u\Vert\,\Vert v\Vert\cos\theta$. It is the shadow of one arrow on the other, scaled. Turn $\vec v$ around $\vec u$ and watch the number cross zero exactly when the arrows stand perpendicular — that single zero powers half of geometry.
The shadow
u · v—
angle θ—
the shadow’s length—
verdict—
camellia dashes: the shadow of v on u — it flips sign at 90°
05
Thales
Slide a parallel through a triangle: every ratio it cuts is the same ratio.
A line parallel to the base slices the two other sides. Thalès’ theorem: it slices them in the same proportion — and the little triangle is the big one, shrunk by exactly that factor. One slider, three equal ratios, no exceptions.
The blade
k where the parallel cuts0.5
AM / AB—
AN / AC—
MN / BC—
verdict—
the emerald blade stays parallel — and the three ratios stay one
Épreuves the page checks by itself—
Six commissions in the geometer’s hand — and an endless examen. The page measures; the compass never lies.
06
Pitfalls
Six ways geometry catches the careless.
Pythagore is conditionalno right angle, no theorem. Use the law of cosines and let $\cos C$ pay the difference.
A vector is not a pointit has length and direction, no address. Two equal vectors can live on opposite sides of the plane.
u · v = 0 needs both aliveperpendicularity is the interesting reason — but the zero vector kills the product too.
Thalès needs parallelsno parallel, no proportion. The theorem’s converse is the cheapest parallelism test there is.
Angles wear units$\sin 30°$ and $\sin 30$ differ wildly. Geometry loves degrees, analysis loves radians — declare your side.
Area scales by k²shrink a triangle by $k$ and its area shrinks by $k^2$. Thalès’ little triangle at $k=\tfrac12$ holds a quarter of the area, not half.