Properties of Figures: Practice Problems and Methods

Triangle properties and centers

The centroid G divides each median $2 : 1$. The incenter I is where the angle bisectors meet, and $\angle BIC = 90^\circ + \frac{1}{2}\angle A$. The circumcenter O is the center of the circumcircle; for an acute triangle $\angle BOC = 2\angle A$. At the orthocenter H, $\angle BHC = 180^\circ - \angle A$.

Ceva's and Menelaus' theorems

For Menelaus, go around the triangle "vertex → point → vertex → …" multiplying ratios. The main decision is which triangle and which line to use. For area ratios, use base ratios when heights are equal, and products of sides when an angle is shared.

Circles, the power of a point, constructions

An inscribed angle is half the central angle on the same arc. Opposite angles of a cyclic quadrilateral add to $180^\circ$. The angle between a tangent and a chord equals the inscribed angle on the arc inside it (tangent-chord theorem).

Two circles are classified by comparing the distance between centers $d$ with the sum and difference of the radii. Constructions combine perpendicular bisectors, angle bisectors and parallel lines, for example to divide a segment or to construct a length $\sqrt{ab}$. Polyhedra satisfy Euler's formula $v - e + f = 2$.