Lopside + Upside + Downside = Golden Side

Geometry Level 4

In the regular pentagon, circles of radii r 1 r_1 and r 2 r_2 are each tangent to two sides and also the segment formed by the vertex and the opposite midpoint.

If the ratio of r 1 r_1 to r 2 r_2 is R R , input ⌊ 1 0 5 R ⌋ \lfloor 10^5 R \rfloor as your answer.


The answer is 128407.

Michael Huang by Michael Huang , 29, USA

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1 solution

Chew-Seong Cheong
Jan 30, 2021

Let the side length of the regular pentagon be 1 1 . Then we note that:

{ r 1 cot ⁡ 2 7 ∘ + r 1 cot ⁡ 5 4 ∘ = 1 ⟹ r 1 = 1 cot ⁡ 2 7 ∘ + cot ⁡ 5 4 ∘ r 2 + r 2 cot ⁡ 5 4 ∘ = 1 2 ⟹ r 2 = 1 2 + 2 cot ⁡ 5 4 ∘ \begin{cases} r_1 \cot 27^\circ + r_1 \cot 54^\circ = 1 & \implies r_1 = \dfrac 1{\cot 27^\circ + \cot 54^\circ} \\ r_2 + r_2 \cot 54^\circ = \dfrac 12 & \implies r_2 = \dfrac 1{2+2\cot 54^\circ} \end{cases}

⟹ R = r 1 r 2 = 2 + 2 cot ⁡ 5 4 ∘ cot ⁡ 2 7 ∘ + cot ⁡ 5 4 ∘ ≈ 1.28407904384 ⟹ ⌊ 1 0 5 R ⌋ = 128407 \begin{aligned} \implies R & = \frac {r_1}{r_2} = \frac {2+2\cot 54^\circ}{\cot 27^\circ + \cot 54^\circ} \approx 1.28407904384 \\ \implies \lfloor 10^5 R \rfloor & = \boxed {128407} \end{aligned}

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