## Thursday, April 18, 2019

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

Details: Click on the figure below.

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

2. Branav Kannan

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

Peter Tran