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Proof that ab = mn

LNM = β = 90°

Line Segment DE = X = 2

X / R = 1.15470053838

Blue Area = 21 Square Units

Easy way

Segment X = CD = 10

Radius R₂ = 8

Angle BCD = X = 10°

Yellow = 0.380784979457

Blue Fraction = 2 / 9

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Angle CAM = α = 15°

Yellow = 64 Square Units

CD = 1.50597117915

Blue Fraction = 1 / 6

Red / Blue = 11 / 9

Perimeter of Blue Area = 32

Blue Fraction = 0.2375 = 19 / 80

Common External Tangent

Area of Square = 98.0643797503

𝛼 = 60°

Top 10.342

Ratio of a : b = 1 : 1

Line Segment K = 6

Total Area = 25 Square Units

Purple : Green = 1 : 1

Angle BAC = 50°

Blue / Total = 0.52072594216

Square Area = 90 Sq. Units.

Blue / Total = 11 / 108

Angle ABC = 52.5

Yellow Area = 2 Square Units

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Baseline AD = 24

Shaded = 3 𝜋 = 9.424778

Red - Green = 22.5

Ratio of a / b = 2

Shaded = 7.79422863406

Angle ABD = X = 15°

Yellow Area = 2 Square Units

BCD = X = 20°

Ratio of A : B : C = 3 : 2 : 1

Shaded Area = a²

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Blue = | OP | ⋅ | KN | / 2

Proofs Without Words

Segment b = 9.79795897113

α / β = 0.5 = 1 / 2

Black Area = 5.04234789804

Line Segment DE = 12

Radius R = 2.57142857143

Maximal Blue Fraction

Angle BAC = 90°

Ratio of Yellow / Green = 5