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Triangle Centers

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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