Jump to content
BrainDen.com - Brain Teasers
  • 0

A triangle defined by cevians


Go to solution Solved by bonanova,

Question

12 answers to this question

Recommended Posts

  • 0
  • Solution

I gave up on finding a rigorous proof.

 

My 1/7 answer was based on a carefully drawn figure and a guess that the ratio might be integral.

I thought I could do the analysis for an equilateral triangle but I gave up even on that case.

Any triangle can be affinely transformed (and thus preserve relative areas) into equilateral.

 

Instead, I found a formula (without derivation) in some obscure online class notes.

 

If cevians cut opposite sides into fractions p and q you get the fractional area of the inside triangle as:

 

Area (inside triangle) / Area (large triangle) = { p1p2p3 - q1q2q3 }2 / { (p1 + q2q3) (p2 + q3q1) (p3 + q1q2) }

It's not difficult to plug in 1/3 and 2/3 respectively for all the p and q to get 1/7 as a ratio.

 

Interestingly, there is one other set of p and q values that leads an integer ratio.

Side ratios of 2/5 and 3/5 lead, surprisingly, to an area ratio of 1/19.

Link to post
Share on other sites
  • 0

Suppose ABC has area of 1.

 

Cevians divide areas in proportion to lengths on the opposite side.

That is given as 1/3, so the cevians define three surrounding triangles that have area of 1/3.

However, they overlap each other near the vertices.

The area of the inner triangle is thus given by the sum of these overlaps.

 

The cevians themselves are divided into segments that appear to have ratios of 1:3:3.

If so, then 1/7 of each surrounding triangle is overlapped.

The central triangle thus has area of 3 x (1/3) x (1/7) = 1/7.

 

There must be an obvious way to show the 1:3:3 cevian proportions, but I don't see it.

Link to post
Share on other sites
  • 0

I assumed each vertex was the intersection of a 1/3 side and a 2/3 side.
The figure makes the most sense that way.

 

Edit::

 

I see now that the cevians are divided into 3/7 3/7 and 1/7 segments.

It's too verbose a process without a figure, which I may construct later,

but I'm convinced of the 1/7 for the area of the inner triangle.

Edited by bonanova
Link to post
Share on other sites
  • 0

Hmmm 1/19 is what I found.

I gave up on finding a rigorous proof.

 

My 1/7 answer was based on a carefully drawn figure and a guess that the ratio might be integral.

I thought I could do the analysis for an equilateral triangle but I gave up even on that case.

Any triangle can be affinely transformed (and thus preserve relative areas) into equilateral.

 

Instead, I found a formula (without derivation) in some obscure online class notes.

 

If cevians cut opposite sides into fractions p and q you get the fractional area of the inside triangle as:

 

Area (inside triangle) / Area (large triangle) = { p1p2p3 - q1q2q3 }2 / { (p1 + q2q3) (p2 + q3q1) (p3 + q1q2) }

It's not difficult to plug in 1/3 and 2/3 respectively for all the p and q to get 1/7 as a ratio.

 

Interestingly, there is one other set of p and q values that leads an integer ratio.

Side ratios of 2/5 and 3/5 lead, surprisingly, to an area ratio of 1/19.

Link to post
Share on other sites
  • 0

Hmmm 1/19 is what I found.

 

 

A rough sketch for 1:2 split rules that out.

Since 1/19 is the result for a 2:3 split, maybe there's a slip in how the ratio was used in the calculations.

 

Try using a 1:1 ratio, and see if that calculates 0 area.

 

You've done more analysis than I have. :thumbsup:

Link to post
Share on other sites
  • 0

another proof

 

 

post-9659-0-20533500-1428011042_thumb.pn

 

 

1) draw 3 additional lines parallel to each cevian passing through the remaining vertices of 2 triangles.

2) show that 6 other shaded triangles are mirror images of the green triangle and therefore have the same area

3) show that every part of a shaded triangle that lies outside of the big triangle has an equal unshaded part inside the big triangle. Midpoints of big triangle's sides are points of symmetry.

Edited by k-man
  • Upvote 1
Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    No registered users viewing this page.

×
×
  • Create New...