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?

Hints/Solution: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?

Comments:Is there a

geometricinterpretation ofs(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.

Extensions/Variations:

Converse. If your conclusion was given, could you conclude thats(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 quadrilateralif and only ifangle C is a right angle.Go toCyclic Quadrilateral Extension.

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

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