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3.2 — Triangles, Similarity and Pythagoras
The triangle is the only polygon that cannot be deformed without changing the length of a side. Push on a square and it collapses into a rhombus; push on a triangle and nothing happens. That single fact is why bridges, roof trusses, cranes, bicycle frames and electricity pylons are built from triangles, and it is also why triangles are the atom of geometry — every polygon can be cut into them, and every measurement in the physical world can be reduced to measuring one.
1. Congruence: when two triangles are the same
Two figures are congruent when one can be picked up and placed exactly on the other — same shape, same size. For triangles, you do not need to check all six measurements (three sides, three angles). Four combinations of three are enough, and knowing which four is the point.
SSS — three sides equal. The triangle is rigid, so the sides fix everything.
SAS — two sides and the angle between them. The angle fixes how the two sides splay apart, and the third side is then determined.
ASA — two angles and the side between them. The two angles fix the third (they must sum to 180°), and the side fixes the scale.
RHS — in right-angled triangles, the right angle, the hypotenuse and one other side.
And the one that is not enough: SSA. Two sides and an angle not between them can produce two different triangles. Draw the known angle, lay the first side along one arm, then swing the second side like a compass — the arc can cut the other arm in two places. This is called the ambiguous case, and it reappears in Chapter 3.5 as the ambiguous case of the sine rule. It is worth knowing that it exists, because it is the one place where "I have three measurements" does not settle the question.
AAA is also not enough, but it fails differently, and that failure is so useful it gets its own name.
2. Similarity: same shape, different size
Two triangles are similar when their angles are equal. They then have the same shape but possibly different sizes, and here is the consequence that does all the work:
\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k
All corresponding sides are in the same ratio. One triangle is a scaled copy of the other, by a factor k.
Two facts follow that people constantly get wrong:
- Lengths scale by k.
- Areas scale by k^2.
- Volumes scale by k^3.
Double every dimension of a shape and you get four times the area and eight times the volume. This is not a triangle fact; it is a fact about scaling anything, and it explains an enormous amount about the physical world.
Why there are no giant insects
An animal scaled up by a factor k has k^3 times the weight, because weight goes with volume. But the strength of its legs depends on their cross-sectional area, which is only k^2 times larger. So the load per unit of bone strength grows as k^3/k^2 = k.
Scale a mouse to the size of an elephant and its legs snap. That is why an elephant's legs are thick pillars while a mouse's are thin sticks, why an ant can carry many times its body weight, and why a mouse survives a fall that kills a horse. Galileo worked this out in 1638 and it is called the square-cube law.
It also explains why small animals lose heat fast (surface area k^2 against volume k^3, so the ratio worsens as things get bigger), why a shrew must eat constantly, and why large data centres are easier to cool per server than small ones.
Similar triangles measure the unmeasurable
Around 600 BCE, Thales measured the height of the Great Pyramid by waiting until his own shadow was as long as he was tall, then measuring the pyramid's shadow. The pyramid, its shadow and the sun's ray form a triangle similar to the one made by Thales, his shadow and the same ray. Equal angles, so equal ratios:
\frac{\text{pyramid height}}{\text{pyramid shadow}} = \frac{\text{his height}}{\text{his shadow}} = 1
The pyramid's height equals its shadow's length, which you can pace out.
This is the whole idea of indirect measurement, and it is still how it is done. A surveyor's total station, a rangefinder, a smartphone's depth estimate and the parallax method astronomers use to measure the distance to nearby stars are all similar triangles. Chapter 3.5 turns the ratios into named functions and makes it systematic.
3. Pythagoras' theorem
In a right-angled triangle, with the two short sides a and b and the hypotenuse c opposite the right angle:
a^2 + b^2 = c^2
The name means "the stretched side" in Greek, from the rope-stretching surveyors who used it. And the statement is genuinely about areas: the square built on the hypotenuse has the same area as the two squares built on the other sides, added together. The algebra is a description of that picture.
Proof 1: the rearrangement
Take a square of side (a+b) and place four copies of the right triangle inside it, each in a corner, rotated a quarter turn from the last. The hole left in the middle is a square of side c (each of its sides is a hypotenuse).
Total area = four triangles + the hole:
(a+b)^2 = 4 \cdot \tfrac{1}{2}ab + c^2
a^2 + 2ab + b^2 = 2ab + c^2
a^2 + b^2 = c^2
The 2ab cancels and the theorem falls out. All it used was the expansion from Chapter 2.1 and the area of a triangle.
Proof 2: by similar triangles
Drop a perpendicular from the right angle to the hypotenuse, splitting it into pieces of length p and q, with p + q = c.
This creates two small triangles, and each is similar to the original — they share an angle with it and both have a right angle, so all three angles match.
From the first small triangle: \frac{a}{c} = \frac{p}{a}, giving a^2 = pc. From the second: \frac{b}{c} = \frac{q}{b}, giving b^2 = qc.
Add them:
a^2 + b^2 = pc + qc = c(p+q) = c \cdot c = c^2
This proof is often considered the most illuminating, because it shows the theorem is really a fact about similarity — about shape being preserved under scaling — rather than a fact about squares.
There are hundreds of published proofs. Euclid's is Proposition 47 of Book I. One is by James Garfield, later President of the United States, who published it in 1876 while a Congressman.
The converse, which is the part that gets used
If a^2 + b^2 = c^2, then the triangle has a right angle. That direction is what makes the theorem practical, and it is what the rope-stretchers used.
Take a loop of rope with 12 equally spaced knots. Pull it into a triangle with sides 3, 4 and 5. Since 9 + 16 = 25, the corner between the 3 and 4 sides is exactly square. No protractor, no instrument, and accurate to whatever your rope allows. Builders still do this, calling it the 3-4-5 method.
Pythagorean triples are whole-number solutions: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29). There are infinitely many, and there is a formula that generates all the primitive ones (those with no common factor): take any two whole numbers m \gt n \gt 0 and set
a = m^2 - n^2, \qquad b = 2mn, \qquad c = m^2 + n^2
With m=2, n=1 you get (3,4,5). With m=3, n=2 you get (5,12,13). Check the identity: (m^2-n^2)^2 + (2mn)^2 = m^4 - 2m^2n^2 + n^4 + 4m^2n^2 = m^4 + 2m^2n^2 + n^4 = (m^2+n^2)^2.
The Babylonian tablet Plimpton 322, from about 1800 BCE, lists fifteen such triples including (119, 120, 169) — which nobody finds by trial and error, so they had a method more than a thousand years before Pythagoras.
Fermat's Last Theorem, in one paragraph
Pythagoras asks for whole-number solutions to a^2 + b^2 = c^2, and there are infinitely many. Around 1637 Fermat scribbled in a book margin that a^n + b^n = c^n has no whole-number solutions at all for any n \gt 2, and that he had a marvellous proof which the margin was too narrow to contain.
It took 358 years. Andrew Wiles proved it in 1994, using machinery that did not exist in Fermat's lifetime and running to over a hundred pages. Fermat almost certainly did not have a valid proof. Chapter 11.4 tells the story.
4. Areas, and one formula that is better than it looks
Triangle: \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}, where the height is measured perpendicular to the chosen base. Why the half: any triangle is exactly half of a parallelogram, obtained by rotating a copy of it 180° and joining.
Parallelogram: base times perpendicular height. Not base times side — slanting a rectangle over keeps the base and sides the same while reducing the height and therefore the area.
Trapezium: \frac{1}{2}(a+b)h, the average of the two parallel sides times the distance between them.
Heron's formula computes a triangle's area from its three sides alone, with no height needed. Let s = \frac{a+b+c}{2}, the semi-perimeter. Then:
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
For a (3,4,5) triangle: s = 6, so the area is \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = \sqrt{36} = 6. Which matches \frac{1}{2}\times3\times4 = 6.
This matters practically because in the field you can measure three distances far more easily than a perpendicular height. It is how a surveyor finds the area of an irregular plot: break it into triangles, measure the sides, apply Heron to each.
5. The centres of a triangle
A triangle has several distinguished points, each defined by three lines that surprisingly all meet at once. That they meet is a theorem in each case, not an observation.
Centroid — where the three medians meet (a median joins a vertex to the midpoint of the opposite side). It is the triangle's centre of mass: cut the triangle out of card and it balances on a pin at the centroid. It divides each median in a 2:1 ratio.
Circumcentre — where the perpendicular bisectors of the sides meet. It is equidistant from all three vertices, so it is the centre of the unique circle passing through all three. Any three points not in a line determine exactly one circle, and this is how you find it.
Incentre — where the angle bisectors meet. Equidistant from all three sides, so it is the centre of the largest circle that fits inside.
Orthocentre — where the three altitudes meet (an altitude is a perpendicular from a vertex to the opposite side).
Euler noticed in 1765 that the centroid, circumcentre and orthocentre always lie on one straight line, now called the Euler line, with the centroid exactly one third of the way from the circumcentre to the orthocentre. There is no obvious reason this should be true, and it is one of the small pleasures of geometry that it is.
The circumcentre has a direct modern use: trilateration, the technique behind GPS. Knowing your distance from three known points fixes your position, because the three circles meet at one point. Volume III, 8.2 covers the satellite version, which does the same thing in three dimensions with spheres.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Pythagoras
a^2 + b^2 = c^2 \qquad \text{for a right-angled triangle with hypotenuse } c
The proof by rearrangement. Take four copies of the triangle and place them inside a square of side a+b, arranged so their hypotenuses form a tilted square in the middle.
The big square has area (a+b)^2. It contains four triangles, each of area \frac12 ab, plus the tilted square of area c^2:
(a+b)^2 = 4\left(\tfrac12 ab\right) + c^2
Expand the left side using the identity from 2.1 — what algebra is:
a^2 + 2ab + b^2 = 2ab + c^2
Subtract 2ab from both sides:
a^2 + b^2 = c^2
No measurement, no approximation — just the same area counted two ways.
The converse is also true and is what makes the formula useful for building: if a^2+b^2 = c^2 then the angle opposite c is a right angle. This is why a 3-4-5 triangle marked out with string squares a foundation, a technique older than the Pythagoreans.
6. Where this shows up in your life
Every building, bridge and pylon. Triangulation for rigidity; the diagonal brace in a wooden frame is there to stop the rectangle collapsing.
Every distance your phone estimates. Camera autofocus, augmented reality, portrait-mode depth — similar triangles and parallax.
Every screen diagonal. A "27-inch monitor" is the hypotenuse. A 16:9 screen with a 27-inch diagonal is 23.5 by 13.2 inches, computed from a^2 + b^2 = c^2 with a/b = 16/9.
Every straight-line distance calculation in software. The distance between two points is Pythagoras, and Volume I, 4.7's pathfinding heuristics use it directly.
Every carpenter's square corner. The 3-4-5 rope trick, four thousand years old and still the fastest way to check a right angle without an instrument.
Every scaled model, drug dose by body weight and cooking time by joint size. The square-cube law of Section 2, which is why doubling a recipe does not mean doubling the cooking time.
Straight-sided figures are now handled. The circle is a different kind of object — no sides, no corners, and a ratio in it that is not a fraction. That is next.