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In triangle $ \bigtriangleup ABC$, the known sides are: $AB=5$, $BC=6$ and $AC=7$. A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively. Segment KL As to the incircle of the triangle ABC.

Please find the length of the segment $KL$, and show me how to.

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A picture would help. –  Gigili Jun 7 '12 at 6:59
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Proofreading would help even more. Is $S$ the same as $C$? Crosses what straight line? What does it mean to say that $L$ is a "non-vertex angle"? What does "respectively" mean is this context? Sorry, this question makes no sense at all. Please think about what you really mwean to ask, and edit the question accordingly. –  Gerry Myerson Jun 7 '12 at 7:15
    
Much better. Almost there. What does "Segment KL As to the incircle..." mean? –  Gerry Myerson Jun 7 '12 at 11:01

1 Answer 1

enter image description here

1. For any $\triangle ABC$ in which

A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is non-vertex angle, respectively

we have that $\triangle ABC \sim \triangle BKL$. Indeed, $AKLC$ is inscribed quadrilateral $\Rightarrow \angle AKL+\angle ACL=\pi$, but also we have $\angle AKL+\angle BKL=\pi \Rightarrow \angle ACL=\angle BKL$; $\angle KBL = \angle ABC$.

As $\triangle ABC \sim \triangle BKL \Rightarrow |BK|=6x, \ |BL|=5x, |KL|=7x$. So we should find $x$ from additional condition of this problem.

2. I guess, that by

Segment KL As to the incircle of the triangle ABC

TS means following: For inscribe circle of $\triangle ABC$ - $KL$ is tangent line.

enter image description here

In this case we can write, that $|KL|+|AC|=|AK|+|LC| \ \mathbf{^{*)}} \Rightarrow$

$\Rightarrow 7x+7=(5-6x) + (6 - 5x)=11-11x \Rightarrow$

$\Rightarrow 18x=4 \Rightarrow x=\frac{2}{9}$.

$|KL|=7x=\frac{14}{9}$

3. $\mathbf{^{*)}}$ As $AKLC$ is tangent quadrilateral thats why |KL|+|AC|=|AK|+|LC|. (See picture).

enter image description here

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