Geometry: Triangles (Properties & Centers)

• Sum of Angles: The sum of all interior angles is always 180°.
• Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• Triangle Inequality: The sum of any two sides must be strictly greater than the third side. The difference of any two sides must be strictly less than the third side.

These four centers are high-frequency targets in competitive exams:

• Centroid (G): The intersection of the Medians. A median connects a vertex to the midpoint of the opposite side. Property: The centroid divides each median in the ratio 2:1.
• Incenter (I): The intersection of the Angle Bisectors. It is the center of the incircle and is equidistant from all three sides. Property: Angle at incenter $\angle BIC = 90^\circ + \angle A / 2$.
• Circumcenter (O): The intersection of the Perpendicular Bisectors of the sides. Equidistant from all three vertices.
• Orthocenter (H): The intersection of the Altitudes (Heights). Property: Angle at orthocenter $\angle BHC = 180^\circ - \angle A$.

If $\triangle ABC \sim \triangle PQR$, then:

• Corresponding angles are equal.
• Corresponding sides are proportional: $AB/PQ = BC/QR = AC/PR$.
• The Square Rule: The ratio of their Areas equals the square of the ratio of their corresponding sides. $\text{Area}(ABC)/\text{Area}(PQR) = (AB/PQ)^2$.

A line segment connecting the midpoints of any two sides of a triangle is parallel to the third side and is exactly half its length. This simple rule solves many complex area-ratio problems.