Geometric Art: Typography, iPad

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## Thursday, April 18, 2019

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Geometry Problem 1081: Equilateral Triangle, Inscribed Circle, Inradius, Tangent Circles, Radius, Tangent Line, Sangaku Japanese Problem

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Draw line p parallel to BC passing through the centre O of the topmost yellow circle.

ReplyDeleteDraw altitude q through A

Draw altitude r through B

Name the intersection of line q and line r as X and the intersection of line p and line q as Y and obtain 30-60-90 triangle XOY

XY = b + 2c

OX = a + b

Since OX = 2XY (30-60-90 tri)

a + b = 2(b + 2c)

a + b = 2b + 4c

4c = a - b -------------------------- (I)

r = 4c + 3b ------------------------ (II)

2a + b = 5b + 4c

2a = 4b + 4c ---------------------- (III)

In (I), (II)

r = 4c + 3b

= a - b + 3b

r = a + 2b

2b = r – a

b = (r – a)/2

In (I), (III)

2a = 4b + 4c

2a = 4b + a – b

a = 3b

Substitute b as b = (r – a)/2in a = 3b,

a = 3b

a = 3 (r – a)/2

2a = 3r – 3a

5a = 3r

a = 3r/5

Substitute a as 3r/5 in a = 3b,

a = 3b

3r/5 = 3b

b = r/5

Substitute b as r/5 in equation (II),

r = 4c + 3b

r = 4c + 3r/5

4c = r – 3r/5

c = (2r/5)1/4

c = r/10

Branav Kannan

ReplyDeleteYou may need to prove that white circles tangent to yellow and green circles also

Peter Tran