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.

## Comments

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TopNewestFor 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 · 2 years ago

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