Triangle Centers

Triangle Centers - Problem Solving


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

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

Triangle ABCABC has incenter II. Let the incircle be tangential to sides AB,BC,AB, BC, and CACA at points F,D,F,D, and E,E, respectively. If the lengths of BC,CA,BC, CA, and ABAB are 19,22, 19, 22, and 23, 23, respectively, what is the length of AF AF ?

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

ABCABC is an acute angle triangle with points DD and EE on BCBC and ACAC, respectively, such that BEBE and ADAD are altitudes. ADAD and BEBE intersect at HH. If BCA=42\angle BCA = 42 ^\circ and EBA=2DAB\angle EBA = 2 \angle DAB , what is the measure of ABC\angle ABC (in degrees)?

Details and assumptions:
- HH is also known as the orthocenter of the triangle, which is the intersection point of all 3 altitudes.


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