Three little circles

Geometry Level 4

As shown in the diagram below, three overlapping circles with distinct radii define six points. The lengths of 5 line segments (in blue) connecting the points are known to us: 4, 6, 8, 3, 5.

What is the length of the remaining segment (in yellow), to 3 decimal places?

The answer is 8.889.

3 solutions

Vishwash Kumar Γξω
Jun 29, 2017

A nice proof of Haruki's Theorem, for those who are not familiar with it.

Mark Hennings - 3 years, 10 months ago

This solution needs to be rotated because I need to turn my head to read the solution.

A Former Brilliant Member - 3 years, 10 months ago

Hey! There is a mistake in the proof.In the second last line you proved BP/DP =bf/ce. But in the last line you have used BP/DP=ce/bf. I first got confused after reading the last line.

Kunal Kundwani - 3 years, 10 months ago

Oh yes I see.

Vishwash Kumar ΓΞΩ - 3 years, 10 months ago

Without knowing Haruki's Theorem, but using Geogebra, I tried to draw a picture of the 3 circles with the right lengths and could not find any way to do it. I guess there is a matter of calculating the radii. Even with the answer, I am unable to make it. Could any one draw the 3 circles with the right scales ? Thanks a lot.

Gerard Boileau - 3 years, 3 months ago

A Former Brilliant Member
Jul 17, 2017

By the Haruki's Theorem , we have

$6(3)(x)=8(5)(4)$ $\color{#D61F06}\large \implies$ $18x=160$ $\color{#D61F06}\large \implies$ $x=8.889$

Jon Haussmann
Jun 29, 2017

This is an instance of Haruki's Theorem:

https://en.wikipedia.org/wiki/Haruki%27s_Theorem

http://mathworld.wolfram.com/HarukisTheorem.html

Three little circles

The answer is 8.889.

3 solutions

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