Let there be a triangle $\triangle ABC$ with incenter $I$. Incircle touches $\overline{BC}$ at $D$. Then a perpendicular is drawn to $\overline{BC}$ at $D$, which cuts the in circle at $E$. $\overline{AE}$ extended intersects $\overline{BC}$ at $F$.

Prove that the ex-circle of triangle $\triangle ABC$, touching $\overline{BC}$ passes through $F$.

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Please I need help to solve this. Also I'd like to have the properties of in-circle, ex-circle, orthocentre cleared (explained).

  • 1
    $\begingroup$ You can consider $\Delta ABC$ as a triangle in $\mathbb{R^2}$ with the property O is mid point of $BC$ and Prove $AB+BF$ is semi perimeter of $\Delta ABC$. $\endgroup$ – R Salimi Aug 12 '13 at 13:56
  • $\begingroup$ @RSalimi can u explain $\endgroup$ – maths lover Aug 12 '13 at 16:27

Let's draw the line $\alpha$ perpendicular to $DE$, passing through $E$
Suppose that $\alpha$ cut $AB$, $AC$ at $X$, $Y$

$(I)$ is the ex-circle of $\triangle AXY$, where $E$ is the tangency point
$A$ is the similarity center of $\triangle AXY$ and $\triangle ABC$
Homothety(center=$A$) send $E$ to the tangency point of $\triangle ABC$ and its excircle

  • $\begingroup$ Prove that the ex-circle of triangle △ABC, touching BC passes through F. $\endgroup$ – maths lover Aug 13 '13 at 2:11
  • $\begingroup$ The proof is already done. $\endgroup$ – chloe_shi Aug 13 '13 at 4:34
  • $\begingroup$ where is it done? $\endgroup$ – maths lover Aug 13 '13 at 17:09
  • $\begingroup$ in last sentence $\endgroup$ – chloe_shi Aug 13 '13 at 18:22

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