# 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.

Note by Samuraiwarm Tsunayoshi
3 years, 2 months ago

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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.....

- 3 years, 1 month ago

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