Easy to see, but so hard to prove! (Help meh)

Use ANY theorem you want!

1.) Let $O$ be a center of inscribed circle in $\square ABCD$ . From point $A$ , draw a perpendicular line to $\overline{AB}$ intersect an extension of $\overline{BO}$ at $M$ , and from point $A$ , draw a perpendicular line to $\overline{AD}$ intersect an extension of $\overline{DO}$ at $N$ . Prove that $\overline{MN}$ is perpendicular to $\overline{CA}$ .

2.) From figure above, $\square AFDC$ has a circle with center $O$ inscribed in it. Prove that

The circumcircle of $\triangle EAF$ , $\triangle ECD$ , $\triangle BDF$ , $\triangle BCA$ , intersect at a single point, say $M$ .
Points $B,M,E$ are collinear.
$\square MDOA$ has a circumcircle.

#Geometry #HelpMe!

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Comments

For 2) I realized that point $M$ is Miquel point of $\triangle BDF$ with points $C, E, A$ on sides $\overline{BD},\overline{DF},\overline{FB}$ respectively.

To prove that it lies on circumcircle of $\triangle BDF$ is unknown.....

Samuraiwarm Tsunayoshi - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$