Posted by sarah A on Sunday, April 11, 2010 at 9:48pm.
The perimeter of a triangle, its area, and the radius of the circle inscribed in the triangle are related in an interesting way. Prove that the radius of the circle times the perimeter of the triangle equals twice the area of the triangle.

geometry  Reiny, Sunday, April 11, 2010 at 10:54pm
Construct "any" triangle. Do not use a special triangle such as an equilateral or rightangled triangle
the inscribed circle will have its centre at the intersection of the angles of the triangle.
Draw a radius to each of the sides, call it r.
It is trivial to see that we have 3 pairs of congruent triangles. Let's call their bases a, b, and c
So the perimeter of the triangle = 2a + 2b + 2c
So the product of the radius and the perimeter is
r(2a + 2b + 2c) = 2r(a+b+c)
Now the area of the whole triangle
= 2(1/2)ar + 2(1/2)br +2(1/2)cr
= ar+br+cr
= r(a+b+c)
All Done!
Answer This Question
Related Questions
 geometry  The radius of the inscribed circle is equal to twice the area of the ...
 Maths  One of the formulas to find the area of a triangle is a = r*s. Where the...
 GEOMETRY CIRCLE  1.An isosceles triangle with each leg measuring 13 cm is ...
 Algebra 2  an equilateral triangle has an inscribed circle with a radius of ...
 GEOMETRY CIRCLES PLEASE  1.An isosceles triangle with each leg measuring 13 cm ...
 GEOMETRY CIRCLES  1. Ab is a chord of a circle with center o and radius 52 cm...
 Geometry about the circles  1. Ab is a chord of a circle with center o and ...
 Geometry about the circles  1. Ab is a chord of a circle with center o and ...
 GEOMETRY CIRCLES PLEASE  HELP ME PLEASE. 1.An isosceles triangle with each leg...
 Math  The radius of the inscribed circle is 5. If the length of the hypotenuse ...
More Related Questions