Triangle Calculator

Fundamental Facts & Definitions

Sides & Vertices: A triangle has three sides, three angles, and three vertices.

Angle Sum: The sum of the interior angles of any triangle is always 180°.

Convex Polygon: All triangles are convex, meaning any line segment connecting two points inside the triangle lies entirely within it.

Triangle Inequality Theorem: The sum of the lengths of any two sides must be greater than the length of the remaining side.

       \( a + b > c \)

       \( a + c > b \)

       \( b + c > a \)

       (This is how you check if three given lengths can form a triangle.)

Triangle Classification

1. By Side Lengths

   Equilateral: All three sides are equal. All three angles are equal to 60°.

   Isosceles: Two sides are equal (the legs). The angles opposite the equal sides are also equal (base angles).

   Scalene: All three sides are of different lengths. All three angles are different.

2. By Interior Angles

   Acute: All three interior angles are less than 90°.

   Right: One interior angle is exactly 90°. The side opposite the right angle is called the hypotenuse, and it is the longest side.

   Obtuse: One interior angle is greater than 90°.

Key Theorems & Properties

1. Pythagorean Theorem

   Applies to: Right triangles only.

   Statement: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

   Formula: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

2. Triangle Congruence Theorems

(Proving triangles are identical in shape and size.)

   SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of another.

   SAS (Side-Angle-Side): If two sides and the included angle are congruent to those of another.

   ASA (Angle-Side-Angle): If two angles and the included side are congruent to those of another.

   AAS (Angle-Angle-Side): If two angles and a non-included side are congruent to those of another.

   HL (Hypotenuse-Leg): A special case for right triangles. If the hypotenuse and one leg are congruent.

3. Triangle Similarity Theorems

(Proving triangles have the same shape but different sizes.)

   AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another, the triangles are similar. (This is sufficient because the third angles must also be equal.)

   SSS (Side-Side-Side): If the corresponding sides of two triangles are proportional.

   SAS (Side-Angle-Side): If two sides are proportional and the included angle is congruent.

4. Special Line Segments & Their Points of Concurrency

   Median: A segment from a vertex to the midpoint of the opposite side. The three medians intersect at the centroid.

       The centroid is the center of mass and always lies inside the triangle.

       It divides each median into a 2:1 ratio (the segment from the vertex to the centroid is twice as long).

   Altitude: A perpendicular segment from a vertex to the line containing the opposite side. The three altitudes intersect at the orthocenter.

       In an acute triangle, the orthocenter lies inside.

       In a right triangle, it lies at the vertex of the right angle.

       In an obtuse triangle, it lies outside.

   Perpendicular Bisector: A line that passes through the midpoint of a side and is perpendicular to it. The three perpendicular bisectors intersect at the circumcenter.

       The circumcenter is the center of the circumcircle (a circle that passes through all three vertices).

       It can be inside, outside, or on the hypotenuse of a right triangle.

   Angle Bisector: A segment that divides an angle into two equal angles. The three angle bisectors intersect at the incenter.

       The incenter is the center of the incircle (a circle tangent to all three sides).

       It always lies inside the triangle.

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1. Important Laws & Formulas

1. Law of Sines

   Use: Relates sides and their opposite angles. Useful for AAS, ASA, and SSA (the ambiguous case) triangles.

   Formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \)

       where \( R \) is the radius of the circumcircle.

2. Law of Cosines

   Use: Generalization of the Pythagorean Theorem for any triangle. Useful for SAS and SSS triangles.

   Formula: \( c^2 = a^2 + b^2 – 2ab\,\cos C \)

       If angle \( C \) is 90°, \( \cos 90° = 0 \), and it reduces to the Pythagorean Theorem.

3. Area Formulas

   Standard Formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

   Using Trigonometry (SAS): \( \text{Area} = \frac{1}{2}ab\,\sin C \)

       (Where \( a \) and \( b \) are two sides, and \( C \) is the included angle.)

   Heron’s Formula (SSS): \( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)

       where \( s \) is the semi-perimeter: \( s = \frac{a+b+c}{2} \)

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1. “Famous” or Advanced Triangle Concepts

   Midsegment Theorem: The segment connecting the midpoints of two sides is parallel to the third side and is half its length.

   45-45-90 Triangle (Isosceles Right): The sides are in the ratio \( 1 : 1 : \sqrt{2} \).

   30-60-90 Triangle: The sides are in the ratio \( 1 : \sqrt{3} : 2 \).

   Exterior Angle Theorem: The measure of an exterior angle is equal to the sum of the two remote interior angles.

   Euler Line: In most triangles, the orthocenter, centroid, and circumcenter are collinear (lie on a straight line). The centroid lies between the other two and divides the segment from the orthocenter to the circumcenter in a 2:1 ratio.

   Ambiguous Case (SSA): When given two sides and a non-included angle, it is possible to create zero, one, or two distinct triangles. This is why SSA is not a congruence theorem.