s(s-c) = (s-a)(s-b)

Suppose you are given a triangle with sides of lengths a, b, and c. Let the semiperimeter be defined by


If s(s - c) = (s - a)(s - b), what can you conclude about the triangle?


Expand each side of the equation . . .     Do you have a proof that the triangle must be a right triangle?

Did you use the Pythagoran Theorem?   Or, did you use its converse?


Is there a geometric interpretation of s(s-c) or (s-a)(s-b)?    Let's construct the incircle for triangle ABC and label the respective distances from A, B, and C to the points of tangency as  x, y, and z respectively:


Now it is relative easy to see that

a = y + z

b = x + z

c = x + y

s = x + y + z

x = s - a

y = s - b

z = s - c

By constructing lines parallel to AC and BC we see a figure that is twice the area of triangle ABC.  


Converse. If your conclusion was given, could you conclude that s(s - c) = (s - a)(s - b) ? That is, if the triangle is a right triangle, prove that this equation must hold.


Cyclic Quadrilateral Extension. Construct a segment of length equal to the semiperimeter from the point of tangency D on AB along the perpendicular to AB. Show that AHBI is a cyclic quadrilateral if and only if angle C is a right angle. Go to Cyclic Quadrilateral Extension.






Coxeter, H. S. M., (1989).  Introduction to Geometry (2nd ed.). New York:  Wiley.

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