Similarity and Proportion

Introduction

Two figures are similar if one is an exact scaled copy of the other — same shape, different size. Similarity is one of the most powerful ideas in geometry, underlying trigonometry, map-making, architectural scale models, and countless geometric proofs. This chapter develops the theory of proportion, proves the similar triangle theorems, and introduces the Golden and Silver ratios as naturally occurring proportions.

Key Takeaways

Two triangles are similar if two pairs of angles are equal (AA), or all three pairs of sides are proportional (SSS), or two sides are proportional with the included angle equal (SAS).
A line drawn parallel to one side of a triangle divides the other two sides proportionally.
The perimeters of similar polygons are in the ratio of corresponding sides; their areas are in the ratio of the squares of corresponding sides.
The Golden Ratio φ = (1 + √5) / 2 ≈ 1.618 arises when a line is divided so that the whole is to the larger part as the larger is to the smaller.
The mean proportional between two segments is found by the altitude-on-hypotenuse construction in a right triangle.

What is Similarity?

Similar geometric shapes — Similar figures share the same shape but differ in size — one is a uniform scaling of the other.

Similar figures are figures that have exactly the same shape but may differ in size. One figure can be obtained from the other by uniform scaling (enlarging or reducing), together with any combination of translation, rotation, and reflection.

All circles are similar to each other.
All squares are similar to each other.
All equilateral triangles are similar to each other.
Ellipses are not all similar; rectangles are not all similar; isosceles triangles are not all similar.

Congruent shapes are a special case of similar shapes (identical in both shape and size). Every congruent pair is also similar, but similar shapes are not automatically congruent.

Proportion

Ratio — the comparison of two quantities of the same kind, expressed as a fraction or with a colon.

Proportion — the statement of equality between two ratios:

a : b = c : d, also written a : b :: c : d, or a/b = c/d

Read as: "a is to b as c is to d."

Terms of a Proportion

Means and extremes of a proportion — In a proportion a:b = c:d, the extremes are a and d, and the means are b and c. The product of the means equals the product of the extremes.

Name	Position	Example
Extremes	first and fourth	a and d
Means	second and third	b and c
Antecedents	first and third	a and c
Consequents	second and fourth	b and d

Fourth proportional — the term d in a : b = c : d is the fourth proportional to a, b, and c.

Continued proportion — quantities a, b, c, d, e are in continued proportion if a:b = b:c = c:d = d:e.

If three quantities are in continued proportion (a:b = b:c), then b is the mean proportional between a and c, and c is the third proportional to a and b.

Proportion Theorems

Cross-product rule for proportions — The cross-product rule: in a proportion a/b = c/d, multiplying diagonally gives ad = bc.

Alternation theorem for proportions — Alternation: if a:b = c:d, then a:c = b:d — the means can be swapped to form an equivalent proportion.

In any proportion, the product of the extremes equals the product of the means: ad = bc.
Corollary: The mean proportional between two quantities equals the square root of their product.
Corollary: If two antecedents are equal, the two consequents are equal.
Alternation — if a:b = c:d, then a:c = b:d.
Inversion — if a:b = c:d, then b:a = d:c.
Composition — if a:b = c:d, then (a+b):b = (c+d):d.
Division — if a:b = c:d, then (a−b):b = (c−d):d.
In a series of equal ratios, the sum of all antecedents is to the sum of all consequents as any single antecedent is to its consequent.
Like powers of the terms of a proportion are in proportion.
If three quantities are in continued proportion (a:b = b:c), then a:c = a²:b².

Proportionality in Triangles

A line parallel to the base divides the other two sides proportionally — A line drawn parallel to the base of a triangle divides the other two sides in equal ratios — the Basic Proportionality Theorem.

If a line is drawn through two sides of a triangle parallel to the third side, it divides the two sides proportionally.
Corollary: One side of a triangle is to either of its segments (cut by a line parallel to the base) as the third side is to its corresponding segment.
Corollary: Three or more parallel lines cut proportional intercepts on any two transversals.
If a line divides two sides of a triangle proportionally from the vertex, it is parallel to the third side.

Dividing a Line into Segments

A line divided internally and externally — A segment divided internally (point between the endpoints) and externally (point on the extension) in the same ratio forms a harmonic range.

A line AB is divided internally at a point O between A and B (segments AO and OB). It is divided externally at a point O in the prolongation beyond one end.

Harmonic division — a line is divided harmonically when it is divided internally and externally into segments having the same ratio. The four points form a harmonic range.

Bisector Theorems

Interior angle bisector dividing the opposite side — The interior angle bisector divides the opposite side internally in the ratio of the two adjacent sides.

Exterior angle bisector dividing the opposite side externally — The exterior angle bisector divides the opposite side externally in the same ratio, completing the harmonic division.

The bisector of an interior angle of a triangle divides the opposite side internally into segments proportional to the adjacent sides.
The bisector of an exterior angle at the same vertex divides the opposite side externally in the same ratio.
Corollary: The internal and external bisectors at a vertex divide the opposite side harmonically.

Similar Polygons

Polygons with corresponding angles equal and corresponding sides proportional are similar polygons. Corresponding sides are also called homologous sides.

Corresponding lines in similar triangles — medians, altitudes, and angle bisectors are all in the ratio of similitude.

The ratio of similitude is the ratio of any two corresponding sides.

Similar Triangle Theorems

Similar parallelograms illustrating the ratio of similitude — Similar parallelograms — the ratio of any pair of corresponding sides gives the ratio of similitude for the whole figure.

Two similar triangles with labelled corresponding sides — Two similar triangles with proportional corresponding sides — the ratio of similitude determines all linear measurements.

Two mutually equiangular triangles are similar.
Two triangles are similar if two angles of one equal two angles of the other.
Two right triangles are similar if an acute angle of one equals an acute angle of the other.
If two triangles have an angle of one equal to an angle of the other and the including sides proportional, they are similar.
If two triangles have their sides respectively proportional, they are similar.
Two triangles with sides respectively parallel, or respectively perpendicular, are similar.

Similar Polygon Theorems

The perimeters of two similar polygons have the same ratio as any two corresponding sides.
Similar polygons can be separated into the same number of triangles, similar each to each and similarly placed.

Right Triangle Corollaries

Altitude drawn from the right angle to the hypotenuse — The altitude from the right-angle vertex to the hypotenuse creates two smaller triangles, each similar to the original — giving mean proportional relationships.

If in a right triangle a perpendicular is drawn from the right-angle vertex to the hypotenuse: - The two smaller triangles are similar to each other and to the original triangle. - The perpendicular is the mean proportional between the two segments of the hypotenuse. - Each leg is the mean proportional between the hypotenuse and the adjacent segment.

Circle Corollaries

Intersecting chords inside a circle showing equal products of segments — Intersecting chords theorem: when two chords cross inside a circle, the product of each chord's segments are equal — a direct consequence of similar triangles.

The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter.
If two chords intersect within a circle, the product of the segments of one equals the product of the segments of the other.
If from an external point a secant and a tangent are drawn to a circle, the tangent is the mean proportional between the whole secant and its external segment.
Corollary: From a fixed external point, the product of any secant and its external segment is constant.

Secant drawn to a circle from an external point — A secant from an external point: the tangent length is the mean proportional between the full secant length and its external segment.

The Golden Ratio

If a line is divided so that the whole line is to the larger segment as the larger segment is to the smaller, it is divided in extreme and mean ratio — the Golden Section.

If the shorter segment has length 1, the longer has length φ = (1 + √5) / 2 ≈ 1.618.

Golden Rectangle — a rectangle whose sides are in the ratio 1 : φ. Removing the largest square from a Golden Rectangle leaves a smaller rectangle with the same proportions.

Nested Golden Rectangles — Successive Golden Rectangles formed by removing squares — each remaining rectangle has the same 1:φ proportions, converging on the golden spiral.

Golden Ratio proportions illustrated — The Golden Ratio expressed geometrically: the relationship between line segments demonstrates the self-similar proportion φ = 1 + 1/φ.

Golden Ratio and the pentagon — the diagonal of a regular pentagon divided by its side equals φ.

Golden Ratio in a regular pentagon — The regular pentagon encodes the Golden Ratio: each diagonal is φ times the side length, and the diagonals themselves intersect in the Golden Ratio.

Construction of the Golden Mean

Phi marked on a divided line segment — The Golden Mean marked on a line: the ratio of the full segment to the longer part equals φ ≈ 1.618.

Extended line showing the external Golden Mean point — Extending the line to find the external Golden Section point, giving a ratio of φ² ≈ 2.618 from the far end.

To divide a segment AB in extreme and mean ratio:

Form circles A(B) and B(A) (each centred at one endpoint, radius equal to AB).
Let C and D be the intersections of the two circles.
Extend AB beyond A to meet circle A(B) at E.
Draw circle E(B); let F be the intersection of E(B) and B(A) farther from D.
DF intersects AB at G. Then AG : BG = φ.

The ratio can also be found externally: extending to a point K gives AB = 1 and KA = φ, so KA + AB = φ² ≈ 2.618.

The Silver Ratio

A lesser-known irrational ratio constructed by an analogous method is the Silver Ratio δ_s = 1 + √2 ≈ 2.414.

Two quantities are in the silver ratio when the ratio of the smaller to the larger equals the ratio of the larger to the sum of the smaller and twice the larger.

Silver Rectangle — cutting two squares from a Silver Rectangle leaves a rectangle with the same proportions.

Silver Rectangle with squares removed — Removing two squares from a Silver Rectangle yields a smaller rectangle with identical 1:δ_s proportions — the silver analogue of the Golden Rectangle.

Silver spiral derived from the Silver Rectangle — The silver spiral, generated by successive Silver Rectangles, grows at a rate governed by δ_s ≈ 2.414.

Silver Ratio and the octagon — the silver ratio appears as the ratio of the second diagonal to the side of a regular octagon, just as the golden ratio appears in the regular pentagon.

Silver Ratio in a regular octagon — The regular octagon embeds the Silver Ratio: the second diagonal (spanning three vertices) divided by the side length equals δ_s ≈ 2.414.

Silver ratio star pattern from an octagon — The eight-pointed star derived from the regular octagon, whose proportions are governed throughout by the Silver Ratio.

Classical Euclidean Constructions

Problem 1 — Third Proportional

To find a third proportional c to two given quantities a and b (so that a:b = b:c).

Construction to find a third proportional — Finding the third proportional: two segments are placed at an angle, and a parallel line locates the third segment so that a:b = b:c.

Place AB and AC at any angle. Set BD = AC. Join BC and draw DE through D parallel to BC, meeting AC extended at E. Then AB:AC = AC:CE, so CE is the required third proportional.

Problem 2 — Fourth Proportional

To find a fourth proportional d to three given quantities a, b, c (so that a:b = c:d).

Animated construction to find a fourth proportional — Finding the fourth proportional: three known segments are arranged on two rays from a point, and a parallel line completes the proportion a:b = c:d.

Set DG = a, GE = b, DH = c on two rays from D. Join GH and draw EF through E parallel to GH. Then DG:GE = DH:HF, so HF = d.

Problem 3 — Mean Proportional

To find a mean proportional b between two given quantities a and c (so that a:b = b:c).

Animated construction of the mean proportional using a semicircle — Finding the mean proportional: a semicircle is drawn on the combined length, and the perpendicular from the join point gives the geometric mean b, where a:b = b:c.

Place AB = a and BC = c in a straight line. Describe a semicircle on AC. Erect the perpendicular BD from B. Then BD is the mean proportional between AB and BC — since the angle ADC is a right angle (angle in a semicircle), and BD is the altitude from the right angle to the hypotenuse.

Conclusion

Similarity transforms congruence into a scale-independent relationship, allowing shape to be studied independently of size. The proportionality theorems — from the Basic Proportionality Theorem to the mean proportional construction — provide the algebraic backbone that underlies both trigonometry and the theory of conic sections. It is precisely because similar triangles have proportional sides that the trigonometric ratios can be defined as functions of angle alone.

The next chapter puts these proportional relationships to work in computing unknown sides and angles — Trigonometry.