Triangle Identities

In this section, we consider several identities applicable to triangles.

Law of Sines:  If ABC is a triangle (where a is the side opposite angle A, etc.), then the following identity holds:

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Proof:  In triangle ABC, drop an altitude from point C to side AB (extending AB if necessary).  Let point D be the intersection of the altitude with side AB, and let the length h be the length of CD.

Considering triangles ACD and BCD, we have , from which one proportion follows.  (If angle A is obtuse, then we actually have , and the proportion still follows.)  Dropping an altitude from a second vertex will provide a second proportion and prove the theorem.♦

Alternate Proof:  Triangle ABC can be circumscribed by a circle.  Let O be the center of the circle, and construct segments OA and OB (each having length equal to the radius r of the circle).  Let P be the midpoint of segment AB, and construct segment OP.

Since triangle OAB is isosceles, triangles OPA and OPB are congruent right triangles.  By the Inscribed Angle Theorem of geometry, angle ACB is half the measure of angle AOB.

Now, if angle ACB is acute, then angle ACB is congruent to angle AOP, and we have:

If angle ACB is obtuse, then angle AOB is the external angle, and we have: 

If angle ACB is a right angle, then side c is the diameter, and the same result still holds.  The other ratios can be handled similarly, with the same result.  Thus, the theorem is proved (with the additional information that the ratio is the diameter of the circumscribed circle).♦

Law of Cosines:  If ABC is a triangle (where a is the side opposite angle A), then the following identity holds:

♦

Proof:  We begin with the same pictures as used in the first proof of the Law of Sines.  Let h be the length of CD, and x be the length of AD.  Then and , where the sign depends on whether angle A is acute or obtuse.  By the Pythagorean Theorem, we have:

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Law of Tangents:  If ABC is a triangle (where a is the side opposite angle A, etc.), then the following identity holds:

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Proof:  By the Law of Sines, implies .  Therefore, we have:

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Tangent Product-Sum Theorem:  If ABC is a triangle (but not a right triangle), then the following identity holds:

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Proof:  The identity can be verified by use of the angle sum and difference formulas for the tangent.

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