Properties of Triangles

An exterior angle formed by producing any side of a triangle equals the sum of its two remote interior angles: $\angle ACD = \angle A + \angle B$.

The sum of the three exterior angles formed by producing sides in cyclic order is always $360^\circ$ (four right angles).

If exterior angle is $120^\circ$ and ratio of remote angles is $1:2$: $x + 2x = 120^\circ \implies x = 40^\circ$. Angles are $40^\circ$ and $80^\circ$.

Never include the adjacent interior angle. The theorem applies only to remote interior angles.

Maritime Navigation: The distance-off-by-doubling-the-angle-on-the-bow technique uses exterior angle geometry to find distance to a lighthouse.

A median joins a vertex to the midpoint of the opposite side. The three medians concur at the Centroid ($G$), which always lies inside the triangle.

The centroid divides each median in a $2 : 1$ ratio from vertex to base ($AG = \frac{2}{3}AD$, $GD = \frac{1}{3}AD$).

For median $12\text{ cm}$, $BG = \frac{2}{3} \times 12 = 8\text{ cm}$ and $GE = 4\text{ cm}$.

Medians bisect sides, not angles (except in isosceles/equilateral triangles).

Aerospace Design: Aircraft center-of-gravity calculations for triangular wing surfaces use centroid coordinates.

An altitude is a perpendicular dropped from a vertex to the opposite side. The three altitudes concur at the Orthocenter ($O$).

• Acute triangle: strictly inside.
• Right triangle: exactly at the right-angled vertex.
• Obtuse triangle: strictly outside behind the obtuse vertex.

In $\triangle ABC$ right-angled at $B$, the orthocenter is vertex $B$ itself.

Do not confuse the orthocenter with the centroid. The orthocenter can be outside the triangle.

Structural Engineering: Stress and load vector intersections in triangular transmission towers correspond to orthocentric geometry.

Circumcenter ($S$): Concurrency point of perpendicular bisectors of sides (equidistant from all 3 vertices: $SA=SB=SC=R$).

Incenter ($I$): Concurrency point of internal angle bisectors (equidistant from all 3 sides: radius $r$).

In a right-angled triangle, the circumcenter lies precisely at the midpoint of the hypotenuse ($R = \frac{\text{Hypotenuse}}{2}$).

If hypotenuse is $10\text{ cm}$, circumradius $R = 10 / 2 = 5\text{ cm}$.

The incenter is always inside the triangle, whereas the circumcenter lies outside for obtuse triangles.

Facility Location: Urban planners place regional emergency hubs equidistant from three towns using the circumcenter.

A triangle can be formed if and only if the sum of the lengths of any two sides is strictly greater than the third side: $a+b > c$.

The side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side.

If $\angle A = 70^\circ, \angle B = 60^\circ \implies \angle C = 50^\circ$. The longest side is $BC$ and the shortest is $AB$.

If $a+b=c$, no triangle forms; the points are collinear.

GPS Route Optimization: A straight-line flight is always shorter than a stopover flight because $AC < AB + BC$.