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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

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.....

Log in to reply