Geometry

Triangle Centers - Problem Solving

Let $$A=(2,12)$$, $$B=(10,0)$$ and $$O=(0,0)$$ be the vertices of triangle $$AOB.$$ If $$G$$ is the centroid of the triangle, what are the coordinates of $$G$$ and the area of triangle $$AGB ?$$

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

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 $$AF$$?

$$A=(17,4)$$ is a vertex of triangle $$ABC$$ and $$O=(0,0)$$ is its circumcenter. $$P, Q$$ and $$R$$ are the midpoints of sides $$AB, BC$$ and $$CA,$$ respectively. If the orthocenter of triangle $$PQR$$ is $$H,$$ then what is the equation of line $$AH?$$

$$ABC$$ 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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