# Geometric demonstration involving ex-circles and in-circles of $\triangle ABC$

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

• 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 $\Delta ABC$. – R Salimi Aug 12 '13 at 13:56
• @RSalimi can u explain – 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