Before we begin, let's name the endpoints of the line segments shown inside of â–³ ABC.
Q: the internal point
E: where a line segment intersects AB
F: where a line segment intersects BC
G: where a line segment intersects AC
Looking at the markings on the given diagram, we can see that AQ bisects ∠A and CQ bisects ∠C.
Therefore, we know that these segments are angle bisectors and that they intersect at the triangle's incenter. According to the Concurrency of Angle Bisectors Theorem, the incenter of a triangle is equidistant from the sides of the triangle.
QE = QF = QG
Now that we know that these segments have equal lengths, we can equate the given expressions for QE and QF to solve for x.
