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What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line! See more

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

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Triangle \(ABC\) 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 \( AF \)?

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\(ABC\) 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.

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