Triangles

Main properties

• The sum of the interior angles is A + B + C = 180°.
• Exterior angle = sum of two interior angles: it equals the sum of the two interior angles that are not adjacent to it.
• A side < the sum of the other two sides, and greater than their difference. That is why there is no triangle with sides 3, 4, 8.
• In an isosceles triangle, the base angles are equal.
• The midsegment is parallel to the third side and equal to half of it.
• In a right triangle, the Pythagorean theorem holds: a² + b² = c², where c is the hypotenuse.
• The angle bisector divides the opposite side into segments proportional to the adjacent sides: if the bisector from vertex A meets BC at D, then BD / DC = AB / AC.
• If the similarity ratio is k, the sides differ by a factor of k and the areas differ by a factor of k².

Theorems and area

Name	Formula
Law of cosines	a² = b² + c² − 2bc · cos A
Law of sines	a / sin A = b / sin B = c / sin C = 2R
Area	S = ½ · a · h, S = ½ · a · b · sin C
Heron's formula	S = √(p(p − a)(p − b)(p − c))
Circles	r = S / p, R = abc / (4S)

p = (a + b + c) / 2 is the semi-perimeter. In a right triangle, R = c / 2.

In a right triangle, the square of a leg equals the product of the hypotenuse and the projection of that leg onto the hypotenuse: a² = c · a_c. The square of the altitude: h² = a_c · b_c.

Example: legs AC = 6 and BC = 8 (right angle at C), hypotenuse AB = 10, the altitude to the hypotenuse is 4.8.

Quantity	Value
Legs	6 and 8
Hypotenuse	10
Projections	3.6 and 6.4
Altitude h	4.8

Examples

• Sides 13, 14, 15: p = 21, p − a = 8, p − b = 7, p − c = 6, S = √(21 · 8 · 7 · 6) = √7056 = 84. The altitude to side 14 is h = 2S / 14 = 168 / 14 = 12. r = 84 / 21 = 4, R = 2730 / 336 = 8.125.
• The exterior angle at vertex C is 110°, A = 45°: B = 110° − 45° = 65°. Check: C = 70°, 45° + 65° + 70° = 180°.
• Legs 6 and 8: c = √(6² + 8²) = 10, S = ½ · 6 · 8 = 24 = ½ · 10 · h, h = 4.8. Check: h² = 3.6 · 6.4 = 23.04. The median is 5 (half of the hypotenuse).

Watch out

• In Heron's formula, p is the semi-perimeter, not the whole perimeter.
• The Pythagorean theorem works only in a right triangle.
• An exterior angle and its adjacent interior angle add up to 180°.
• An altitude is perpendicular to the side, not a lateral side.
• Check the triangle inequality before you give your answer.