Advanced Geometry and Coordinate Geometry

GRE geometry at the 160+ level tests knowledge of triangle properties and circle properties that go beyond basic area formulas. For triangles: the exterior angle theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent interior angles — this is frequently used to find angle measures without computing all three interior angles. The triangle inequality theorem states that the sum of any two sides must be greater than the third side and the difference of any two sides must be less than the third side — this determines the range of possible values for an unknown side in QC problems. For circles: the inscribed angle theorem states that an inscribed angle (vertex on the circle) measures half the central angle that subtends the same arc. An inscribed angle in a semicircle is always 90° (this is the corollary most frequently tested). The relationship between a tangent line and a radius: a tangent is always perpendicular to the radius drawn to the point of tangency. Composite figure problems (a shaded region inside one shape and outside another) require computing each area separately and subtracting: shaded area = outer area − inner area. Common composite combinations: circle inscribed in a square (shaded region = square area − circle area); two overlapping circles (overlap area is computed using the sector-minus-triangle method for each circle's contribution to the overlap). Memorize these area formulas precisely: triangle = (1/2)base × height; circle = πr²; sector = (central angle / 360) × πr².