Geometry
# Triangle Centers

$O$ is the circumcenter of $\triangle {ABC}$. If $\angle AOB : \angle BOC : \angle COA = 2:3:4,$ what is the measure of $\angle BAC$?

Point$ABC$ has incenter $I$. Let the incircle be tangential to sides $AB, BC,$ and $CA$ at points $F,D,$ and $E,$ respectively. If the lengths of $BC, CA,$ and $AB$ are $19, 22,$ and $23,$ respectively, what is the length of $AF$?

Triangle$ABC$ is an acute angle triangle with points $D$ and $E$ on $BC$ and $AC$, respectively, such that $BE$ and $AD$ are altitudes. $AD$ and $BE$ intersect at $H$. If $\angle BCA = 42 ^\circ$ and $\angle EBA = 2 \angle DAB$, what is the measure of $\angle ABC$ (in degrees)?

**Details and assumptions:**

- $H$ is also known as the orthocenter of the triangle, which is the intersection point of all 3 altitudes.