Intersections

Geometry Level 2

What is the maximum number of intersection points that can be made with 2 circles and 3 lines?

Note : The above figure show an explicit example with 8 intersection points.


The answer is 17.

Jonathan Hsu by Jonathan Hsu , 19, USA

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6 solutions

Prakhar Dhumas
Apr 5, 2015

5 Helpful 0 Interesting 0 Brilliant 0 Confused

However, couldn't the circles lie on top of each other? Or the lines as well? This would give us infinite intersection points.

Christian Nunez - 5 years, 8 months ago

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I suppose there may need to be a specification that the circles and lines are distinct from each other. But it seemed obvious to me that that wasn't going to be the answer (I mean how do you put infinite as an answer? the word infinite? the symbol?)

A Former Brilliant Member - 5 years, 8 months ago

Yeah I thought the same thing

silvia woods - 5 years, 8 months ago

Excuse me, but if the circles was one above the other. Have they infinite intersection points?

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Interesections are supposed to be singularities so two circles one above the other can't be considered as infinite intersection points.

Alfonso Cabrera - 5 years, 8 months ago
Achille 'Gilles'
Sep 18, 2015

A total of 17 intersections.

Circles crossing twice. That's 2 intersections.

Each lines cross 2 times the 2 circles so 3 lines x 2 circles x 2 times. That's 12 more intersections.

Line 1 cross line 2 and 3, line 2 cross line 3. Finally 3 more intersections.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Steven Lee
Aug 28, 2015

Each line can have two intersections with each circle at maximum, one intersection with each line maximum. So there are a maximum of 3 line-line intersections and 4 × 3 = 12 4 \times 3 = 12 line-circle intersections. Lastly, we add on the two circle-circle intersections, which gives us 17 total intersections.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Thomas Savage
Sep 18, 2015

Most intersections from two circles = 2 -> Add a line into the mix = 6 (+4) -> And another = 11 (+5) -> Bob's your uncle = 17 (+6).

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Long Nguyen
Sep 18, 2015

max intersections of 1 line x 1 line = 1. There are 3 lines -> 3 max of 1 circle x 1 circle = 2. There are 2 circles -> 2

max of 1 circle x 1 line = 2. There are 2 circles x 3 lines -> (2 + 2) * 3 = 12

3 + 2 + 12 = 17

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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