# Find the value of the angle $X$ in the given figure [closed]

In this picture, the curve in the inside of the big outer triangle is actually its incircle. The edges of the triangle inside the incircle are the intersections of the incircle with the outer triangle.

What is the value of angle $X$ in the given figure?

## closed as off-topic by abiessu, Roman83, Watson, David K, user26857Dec 1 '16 at 16:04

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• $x=\frac{180-64}2$ – Roman83 Dec 1 '16 at 14:21
• Information given is incomplete - what relation does the inner triangle have with the outer? – GoodDeeds Dec 1 '16 at 14:22
• Roman83, what is this formula? Could you explain – mather Dec 1 '16 at 14:24
• abiessu, it is a circle. – mather Dec 1 '16 at 14:24
• Okay, so are the corner points of the inner triangle supposed to be on the edges of the outer triangle? It seems that at least one isn't quite lining up that way... – abiessu Dec 1 '16 at 14:26

Let's use the notation as in this picture.

Note that $$\overline{DE} \perp \overline{BC}$$ and $$\overline{DG} \perp \overline{AC}$$. Hence $$\angle GDE= 180-\angle EBG$$. AS the traingle $$\triangle GED$$ is isosceles, we conclude $$\angle DEG= \frac{1}{2}( 180-\angle GDE)= \frac{1}{2} \angle EBG$$ Similarly $$\angle FED= \frac{1}{2} \angle FCE$$

In conclusion $$\angle FEG =\frac{1}{2} (\angle FCE + \angle EBG)$$

• The last line should have $\angle FCE$, not $\angle FEC$. – teadawg1337 Dec 1 '16 at 15:07
• thx, I replaced it now – J.Doe Dec 1 '16 at 15:13
• @J.Doe: how did you make this picture? What software? – Arnaldo Dec 1 '16 at 15:43
• @Arnaldo The software is called GeoGebra – J.Doe Dec 1 '16 at 16:47

Let's call $a,b,c$, ($b$ is opposite to $x$ and $a$ is the up arc) those three arcs on the circle:

$$\frac{a+b-c}{2}=32$$ $$\frac{b+c-a}{2}=84$$

Sum both equations and get $b=32+84$ and then $x=b/2=58$

• What theorem are you using? How did you get 84? – mather Dec 1 '16 at 14:58
• See this: item number 5. mathbitsnotebook.com/Geometry/Circles/CRAngles.html – Arnaldo Dec 1 '16 at 15:32
• $84$ is the thrid angle of the triangle $84=180-(64+32)$. – Arnaldo Dec 1 '16 at 16:17

HINT

It is an in-circle of the given triangle.Formed by corner angle bisectors. Angle chasing can find it.