I Help finding a triangle inside a triangle

aheight

Summary
Finding precise location of triangle in a right triangle with the give conditions
I would like to find the triangle $\bigtriangleup DEF$ in the plot below that is inside the right triangle $\bigtriangleup ABC$ given $\overline{AB}=3, \overline{AC}=4$ with $\overline{BD}=\overline{DE},\overline{AE}=\overline{EF}, \overline{FC}=2\overline{DF}$. However, I'm finding it difficult to actually find precisely in the larger triangle where it is (diagram below is only approx.) . Surely if I let $D=(x_3,y_3),F=(x_2,y_2), E=(x_1,y_1)$ as per the diagram and solve the following six simultaneous equation for the coordinates of D,E, and F inside the larger triangle should do it but when I attempt to numerically solve the equations (in Mathematica), I obtain the empty set. However, the triangle does exist. I was wondering if someone could look at my work and confirm that I have a valid set of equations to find these points? For example, the first equation is obtained by noting:
$x_1^2+y_1^2=z^2$ and $(x_2-x_1)^2+(y_2-y_1)^2=z^2$ and so forth.

Thanks guys.

$$\begin{array}{l} x_1^2+y_1^2=(x_2-x_1)^2+(y_2-y_1)^2 \\ x_3^2+(3-y_3)^2=(x_3-x_1)^2+(y_3-y_1)^2 \\ 4[(x_3-x_2)^2+(y_3-y_2)^2]=(4-x_2)^2+y_2^2 \\ x_2^2+y_2^2=4(x_1^2+y_1^2)\\ 4[x_3^2+(3-y_3)^2]=x_1^2+(3-y_1)^2\\ 9[(x_3-y_2)^2+(y_3-y_2)^2]=(4-x_3)^2+y_3^2 \end{array}$$

Last edited:

Delta2

Homework Helper
Gold Member
I didn't check your equations thorously but I think the reason that mathematica cant solve it, is that the system probably has infinite solutions. I think if you manually enter coordinates for say (x1,y1) and leave x2,y2 and x3,y3 as unknowns, mathematica will be able to solve it.

pbuk

the triangle does exist
How do you know this?
$(x_3−y_2)^2$
Doesn't look right.
[Edit: $\LaTeX$ corrected]

Delta2

Homework Helper
Gold Member
How do you know this?
I think from the scheme is obvious that for this right triangle ABC it exists at least one DEF triangle.

Moreover I think that for any right triangle there exist infinite triangles DEF which "oscillate" around a "central" DEF_0 triangle.

To see this pick any line from B that intersects the side AC at an internal point. Then we can choose randomly a point D on this line and then choose a point E such that BD=DE. Now within a range of angles that the line BD makes with side AB and within a range of magnitude of BD, we can choose point F such AE=EF and E,F are internal to the triangle ABC (they will not be internal for all angles and for all lengths of BD, but within a range of those ,E,F will be internal) so the triangle DEF will be internal.

EDIT: BIG OOPS, i overlooked that we also have the restriction FC=2DF, this might make the triangle non existant...

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aheight

Here's the solution guys (I had an error in the Solve parameters initially). I believe there is only one solution. And here's the Mathematica code I used to solve it (Solve took 30 minutes to find the solution). Also, I confirmed the answer by computing the area via Heron's formula against the known area of 3/5.

Mathematica code:
(* takes about 40 minutes to find the solutions below *)
r1 = Polygon[{{0, 0}, {0, 3}, {4, 0}}];
eqn1 = x1^2 + y1^2 == (x2 - x1)^2 + (y2 - y1)^2;
eqn2 = x3^2 + (3 - y3)^2 == (x3 - x1)^2 + (y3 - y1)^2;
eqn3 = 4 ((x3 - x2)^2 + (y3 - y2)^2) == (x2 - 4)^2 + y2^2;
eqn4 = x2^2 + y2^2 == 4 (x1^2 + y1^2);
eqn5 = 4 (x3^2 + (3 - y3)^2) == x1^2 + (3 - y1)^2;
eqn6 = 9 ((x3 - x2)^2 + (y3 - y2)^2) == (4 - x3)^2 + y3^2;
Solve[{eqn1 && eqn2 && eqn3 && eqn4 && eqn5 && eqn6 &&
Element[Alternatives @@ {{x1, y1}, {x2, y2}, {x3, y3}}, r1]}, {x1,
x2, x3, y1, y2, y3}]
{{x1->4/5,x2->8/5,x3->2/5,y1->3/5,y2->6/5,y3->9/5}}

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Bill Simpson

This takes a fraction of a second to find a solution

Code:
Clear["Global*"]; (*I cannot make it display just and only one grave followed by asterisk*)
a={0,0}; b={0,3}; c={4,0}; d={dx,dy}; e={ex,ey}; f={fx,fy};
Reduce[b-d==d-e && a-e==e-f && f-c==2(d-f), {dx,dy,ex,ey,fx,fy}]
(*which returns*)
dx==2/5 && dy==9/5 && ex==4/5 && ey==3/5 && fx==8/5 && fy==6/5

(*check the result*)
{b-d==d-e, a-e==e-f, f-c==2(d-f)}/.{dx->2/5, dy->9/5, ex->4/5, ey->3/5, fx->8/5, fy->6/5}
(*which returns*)
{True,True,True}`

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"Help finding a triangle inside a triangle"

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