Simple geometry problem -- Find the perimeter of the triangle ABC

[docnet]

Homework Statement

what is the perimeter of the triangle ABC?

Relevant Equations

$\angle A+\angle B + \angle C = \pi$

Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it.

what I know:
The internal angles add up to pi
Let the tangent point between A and B be X
Let the tangent point between B and C be Y
Let the tangent point between C and A be Z
$\overline {AC} = 14$
$\overline {BX} = 16$Things I suspect are true, but have yet to prove:
$\overline {AX} =\overline {AZ}$
$\overline {CZ} =\overline {CY}$
$\overline {BX} =\overline {BY}$

Relations:
$\overline {CY}=14-\overline {AZ}$
$\overline {AX}=\overline {AB}-16$so far I only found three unknowns and only two equations! I think that this is strangely difficult.

 

[ergospherical]

Um, the circle is circumcised?

...anyway, yes, these are true:

Things I suspect are true, but have yet to prove:
$\overline {AX} =\overline {AZ}$
$\overline {CZ} =\overline {CY}$
$\overline {BX} =\overline {BY}$

which you can see by, e.g. considering the right angled triangles $AXP$ and $AZP$ with common hypotenuse, etc.

Now you can see the answer just by looking at the figure, but if you like, denote the three relevant lengths by $a,b,c$ (i.e. $a+c = 14, b=16$) and do it algebraically.

 

[BvU]

@ergo: you rob docnet of the exercise this way!
PF normally restricts help to guiding questions and hints...

 

[docnet]

Um, the circle is circumcised?

...anyway, yes, these are true:

which you can see by, e.g. considering the right angled triangles AXP and AZP with common hypotenuse, etc.

Now you can see the answer just by looking at the figure, but if you like, denote the three relevant lengths by a,b,c (i.e. a+c=14,b=16) and do it algebraically.

aha! the perimeter of the triangle is 16+ 16+14+14 =60. thank you ! it seems obvious now

and I did not realize that it says circumcised. It was probably a typo.

 

[haruspex]

the circle is circumcised?

Else the triangle would have been larger.

 

[valenumr]

Homework Statement:: what is the perimeter of the triangle ABC?
Relevant Equations:: $\angle A+\angle B + \angle C = \pi$

Hello, so I saw this problem on a website while looking up trigonometric identities and trying to solve it.
View attachment 290564
what I know:
The internal angles add up to pi
Let the tangent point between A and B be X
Let the tangent point between B and C be Y
Let the tangent point between C and A be Z
$\overline {AC} = 14$
$\overline {BX} = 16$Things I suspect are true, but have yet to prove:
$\overline {AX} =\overline {AZ}$
$\overline {CZ} =\overline {CY}$
$\overline {BX} =\overline {BY}$

Relations:
$\overline {CY}=14-\overline {AZ}$
$\overline {AX}=\overline {AB}-16$so far I only found three unknowns and only two equations! I think that this is strangely difficult.

Notice the three isoscelese "triangles" created by the circles tangent points? You don't need any trigonometry to solve this only basic algebra and that fact.

 

[haruspex]

Notice the three isoscelese "triangles" created by the circles tangent points? You don't need any trigonometry to solve this only basic algebra and that fact.

As noted in post #2, and used in post #3.

 

[valenumr]

As noted in post #2, and used in post #3.

Yeah, I just answered quickly without reading the thread. A better explanation was also provided proving the equal triangles.