Math Is Fun Forum

johntom

Guest

Calculate the area of a triangle

AB=6, point C is on AB, AC=4, make an equilateral triangle CEG with C as the vertex, connect AE and BG, when AE+BG is the smallest, find the area of triangle CEG.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: Calculate the area of a triangle

Hi johntom...and welcome to the forum!

Unless I don't fully understand the problem (or maybe there some info missing), I think I've found the smallest AE+BG...provided that either AE or BG (depending on the case - see images) can be length zero. If not, then the smallest AE+BG would be an infinitesimally small amount larger than zero!

Also, there seems to be an infinite number of areas possible for triangle CEG (all when AE+BG is smallest), ranging from √3 (see edit below) to 4√3, with the difference between adjacent sizes being immeasurably small.

I've drawn the following images based on AE or BG (as the case may be) having zero length. If zero length is not allowed, the change to the next larger size would be indiscernible (that is, if my conclusions are correct)!

The first image shows triangle CEG, with side CB as the base along the x axis.

Triangle CEG, with vertex C remaining on the x axis, then rotates -120° (ie, anticlockwise) at vertex C. 

The second image shows CEG's position at the end of the -120° rotation, with side EC having become the new base along the x axis:

During the -120° rotation, CEG changes size (first shrinking from its initial 2x2x2, then growing to its final 4x4x4), while maintaining constant smallest AE+BG throughout.

EDIT: The smallest CEG area I've found is 0.7423074889 (rounded to 10 decimal places), when CEG is about 41% into its 120° anticlockwise rotation (see image below).

These images could also have been drawn with CEG under the x axis.

Last edited by phrontister (2024-11-28 21:58:01)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Calculate the area of a triangle

I'm assuming you used Geogebra and moved a point around until you reached a minimum.  I did that using Sketchpad and got this:

I feel that there ought to be an analytical way to do this, so I'm working on that.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: Calculate the area of a triangle

I'm assuming you used Geogebra and moved a point around until you reached a minimum.

True!

I couldn't come up with anything I'd call the proper approach, so I used the 'play and observe' (P&O) technique that works for me sometimes.

So...I parked the small 2x2x2 CEG on CB, and moved G up-left from its starting position at B, along a constant-slope trajectory that produced a constant smallest AE+BG during CEG's 120° anticlockwise rotation at C.

CEG shrank immediately after G departed B, and then grew into 4x4x4 when E reached A.

I found my minimum CEG by moving G slowly along the trajectory line...on a very zoomed-in screen.

Btw, the OP's problem doesn't say anything about the sought area size. We're trying for the smallest atm, and I've mentioned that there's a myriad of sizes between the smallest and largest, but maybe the OP wants the largest? Well, I think the largest is this (as also shown in my second image in post #2):

That would turn this puzzle into quite an easy one!

I'm still trying to work out what you did...

Last edited by phrontister (Yesterday 11:21:35)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: Calculate the area of a triangle

Hi Bob;

I went 'P&O' discovery cruising again this morning, and saw that AEC and CGB are both right angles for minimum CEG.

I zoomed in at Geogebra's maximum level, adjusted CEG's current position a tiny bit, and the standard curved symbol for both angles magically changed into right-angle symbols.

As you can see from my image, I couldn't quite get the angles to actually be 90°, but that was close enough for Geogebra.

Maybe this info can help with what you're working on...

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 245

Re: Calculate the area of a triangle

Although I can't follow you, I just wonder if this problem may have more than one solution.

---

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: Calculate the area of a triangle

Hi Kerim;

Although I can't follow you...

Here's a link to a little video I just made that I hope will help: Triangle area.

It shows one of the paths that equilateral triangle CEG can take. It's the path that I've described in my posts, and as far as I can tell is the only one where the required smallest AE+BG occurs.

...I just wonder if this problem may have more than one solution.

The path I've described maintains the smallest AE+BG for the whole path, not just particular locations of the triangle along the path. That gives innumerable solutions differing in size!

In my post #4, I wrote:

Btw, the OP's problem doesn't say anything about the sought area size. We're trying for the smallest atm, and I've mentioned that there's a myriad of sizes between the smallest and largest, but maybe the OP wants the largest?

I would've thought that, ideally, there'd only be one location for the smallest AE+BG, and that we'd only need to report the area of CEG at that location.

So maybe the OP's wording in post #1 is missing some information we need?

Or...I've simply misunderstood/overlooked/misconstrued something or other!

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 245

Re: Calculate the area of a triangle

Thank you, Phrontister, for the interesting explanation.

---

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,884

Re: Calculate the area of a triangle

Hi Bob;

I took some more measurements from when CEG is smallest, and found these interesting length relationships that don't occur at other CEG locations:

I've had a good look to see if I could use this new info (in addition to the two right angles that only appear then) to improve on my P&O solution technique, but no luck so far.

Also, FB (the line along which G travels) = AE+BG...which it always was, but I hadn't noted it. And I hadn't noted the existence of FB, which is the "constant-slope trajectory" I referred to in post #4.

Btw, the 'F' point in FB is hidden under the 'G' point of the 4x4x4 CEG in the second image in my post #2.

Here's a link to an animation video I made: Triangle area. It uses the image details from post #5.
Note: The two bold-font boxes in the bottom right-hand corner are dynamic, continually updating as CEG moves along its path. The area box changes constantly...but the AE+BG box doesn't, confirming the accuracy of the FB trajectory slope.

Last edited by phrontister (Today 13:12:41)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson